mostly applicable for re-leveling skills as per the allowance of patch 1.9, of limited utility for leveling skills for the first time.

GENERAL ADVICE:

• Have Dawnguard installed. Complete Lost to the Ages. Pick Aetherial Crown as reward. Have the crown store Lover Stone. Superglue it to your character's head.
  • If you know you are going to be concentrating on the stealth/magic/combat skill groups, have your regular standing stone be Thief/Mage/Fighter stone, as applicable.
• DON'T RESET ALL SKILLS AT ONCE. Many skills become infinitely faster and easier to re-level if certain similar skills are high-level or you have certain perks in them. And if you are at a high level, lowering all your vital skills back to 15 at once can make the game extremely hard. Consider skill interplay carefully before resetting them.
• There is no shame in using trainers.
• For re-leveling the schools of magic, be sure to purchase some or all of the high-level spells. Casting Expert-level spells levels the skill faster than Novice-level spells.

MAGE SKILLS

• Before resetting Enchanting, spend all the perk points you can in strengthening enchantments, then make gear of fortify (magic school), rings, necklace, body armor, helm/circlet (latter one optional because of Aetherial Crown) using the best souls/potions/Ahzidal gear (Dragonborn expansion). It's possible to get 100% spell cost reduction, which helps with leveling as generally, the more magicka a spell uses, the more exp it grants.

ILLUSION

• The Dark Brotherhood Forever: use Fury/Frenzy spells on your murder targets to make them aggro the guards and other nearby NPCs and subsequently be killed by them.
  • since fury/frenzy on civilians/citizens is a crime, use Frenzy Rune (Dawnguard expansion added) if you don't have Quiet Casting perk/good Sneak.
• Eschew fast travel, walk everywhere, and keep Muffle active at all times, Invisible too if you have the patience.

CONJURATION

• Sadly, Summon Arvak (Dawnguard expansion added) does NOT seem to level the skill.
• Bound Weapons offer the most EXP when used in open combat rather than stealth combat.
• Buy empty soul gems (levels speech) and cast soul trap on your enemies (traveling overland will ensure many encounters with various white-souled creatures). Use the soul gems (levels enchanting) and either keep or sell the items (levels speech).

DESTRUCTION

• Exp gain is based on the magicka of the spell. Runes (particularly Lightning Rune) have the highest base magicka cost outside of the Wall and Master spells, and grant EXP when they deal their damage. Use stealth, the Quiet Casting perk, and a bit of tactics to take out patrolling enemies in dungeons, Dark Brotherhood Forever targets, or even random city guards if you feel particularly murderous. Note that runes ONLY grant exp when they are triggered, not when they are cast.

RESTORATION

• Healing spells only grant EXP when actually restoring lost health. Lose health in controlled circumstances to be able to heal it back and gain EXP. Fall damage (Markarth is a good jumping ground), the fires of the Aetherium Forge (Dawnguard), and the path to the spike pit (Halted Stream Camp in Whiterun) are decent bets.
• Be a vampire. Cast Circle of Protection. It will always grant max possible exp for the spell, because there will always be an undead (you) inside it upon casting.
• Have Dawnguard installed. Have sided with the Volkihar clan. Talk to one of the castle death hounds (Garmr or CuSith) and make them your follower. Cast Turn Undead spells on the dog repeatedly--it's undead thus is a valid target, but as it is also your follower, it will never run away, turn hostile, or otherwise complicate the powerleveling process.

ALTERATION

• Go to Halted Stream Camp in Whiterun. Find the Transmute spellbook. Learn Transmute, then start visiting iron mines and buying iron ore (NEVER ingots, only ore) from everyone who sells it (blacksmiths, fletchers/armorers, general merchants, Khajiit caravans). Repeatedly cast Transmute to turn your iron ore into silver, then gold, to gain Alteration exp, then turn the ore into jewelry (levels smithing), optionally enchant them (levels enchanting), and then sell them (levels speech).

ENCHANTING

• Clear The Black Star daedric quest. Side with Nelacar to get the Black Star as opposed to Azura's Star. More powerful enchantments level enchanting faster, more powerful enchantments are made with better souls, black souls are the best souls but black gems are rare, the black star is a reusable black soul gem. You can figure it out from here.
• More powerful enchantments level enchanting faster, as do more costly enchants. Fortify Sneak, Archery, Muffle or Carry Weight, and Banish, Paralyze, or Fear are your best bets.
• Recharging enchanted weapons with soul gems slightly increases enchanting, regardless of how badly the weapon actually needs enchanting. Own the Black Star. Kill human opponent with a soul trap weapon. Use trapped soul to recharge weapon. Rinse and repeat. (Be sure to leave a black soul in the Star for when you go to the enchanting table though).

COMBAT SKILLS

• Note that while Archery seems to be grouped with combat skills, you actually use the Thief Stone, not the Fighter Stone, to speed up its leveling.

SMITHING

• Obtain Transmute tome from Halted Stream Camp. Obtain A LOT of iron ore via mining/purchasing (from smiths, armorers, fletchers, general merchants, and khajiit caravans). Transmute iron into gold/silver (levels alteration). Optionally/ideally, have completed No Stone Unturned to have Prowler's Profit. Make jewelry. This requires no perks/previous skill level, and as smithing exp is based on the value of the item made, jewelry increases the skill quickly for a minimum investment. Optionally, enchant them (levels enchanting) and sell (levels speech).
• You will surely acquire many animal pelts while wandering Skyrim like I recommend. When you have many, turn them into leather. While at the tanning rack, turn some of the tanned leather into leather strips until you have as close to an equal number of strips and leathers as you can get. Leather Boots, Bracers, and Helmet total require 5 leathers and 5 strips to create. If you have an unequal amount of strips and leathers, make bracers to reduce strips and helmets to reduce leathers, then craft. Optionally, enchant them (levels enchanting) and sell (levels speech).

HEAVY ARMOR

• Sadly there is no real shortcut for leveling any armor skill. The harder you're hit, the more exp you get.
• My advice for this and light armor, if you want to power-level them, is actually to level in conjunction with block, like so: Get in armor you want to level. Get really good shield. Have Shadowmere. Go about your business on the world map, riding Shadowmere, until you come across some relatively low-damage enemies (bandits and thieves are decent picks as well as wild animals). Dismount from Shadowmere, and sit there with your shield up (Stendarr's Amulet and enchanting the shield with Fortify Block may help) while Shadowmere kicks the enemies to death. Shadowmere is effectively immortal but her damage is not super high, and you've geared yourself to be super stone-wall, so just take the punishment and watch the level ticker go up.
• Mix and match armor pieces, heavy and light, to level both skills at once.

BLOCK

• Have Shadowmere. Ride around until you find an enemy with "safe" DPS. Dismount, put shield up, wait for Shadowmere to kick the enemy to death while you make sure all their attacks hit your shield.

TWO-HANDED

• Like other combat skills, this one is hard to get cheap levels in. If you're desperate for fast leveling, best bet is to get a two-handed weapon with great DPS, and just attack Shadowmere. She's basically immortal and will never turn on you, so you can mash attack for the long, long, LONG process of whacking your way to skill 100.
• Don't forget: No shame in using trainers.

ONE-HANDED

• See Two-Handed.

ARCHERY

• See Two-Handed.

STEALTH SKILLS

• More than any other subset, stealth contains a ton of skills without direct combat efficacy. Since many of them also level quickly and easy, be cautious about gaining too many levels via re-leveling stealth skills unless you are already leveled past the point of enemy scaling.

LIGHT ARMOR

• See Heavy Armor. And remember: Trainers are there to help you.

SNEAK

• The swiftest way to gain Sneak exp is to perform successful sneak attacks. Wear the Dark Brotherhood gloves/handwraps to make your attacks more successful until you re-unlock the multiplier perks.
• Gaining EXP in Sneak is also done by remaining undetected. Eschewing fast travel and roaming overland in Sneak Mode is helpful in this. The illusion spells/effects of Muffle and Invisible are AMAZINGLY helpful in this.
• Avoid resetting Illusion and Sneak at the same time. Illusion's Quiet Casting perk will make using Muffle and Invisible to stay unseen even when right in front of people an absolute breeze.
• When an enemy gives up looking for you, you gain sneak exp. Sneak attack an enemy, and when their friends come looking for you, cast Invisible to become unseen, and wait for them to give up looking for you. Alternately, find a secluded spot and shoot an arrow near enemies. The arrow will alert them, and they will begin looking for you. Wait for them to give up (Invisible may be needed), then try again. This should also work with the Throw Voice shout.

LOCKPICKING

• BE VERY CAREFUL ABOUT RESETTING LOCKPICKING. For some reason, Bethesda programmed Skyrim so that each lock will only offer lockpicking EXP the FIRST time it is picked successfully, and NONE on subsequent attempts. Given how many locks you have probably picked in order to get lockpicking to 100 the first time, there probably aren't many "virgin" locks left. It is still possible to re-level lockpicking, but it takes some doing.
• The best method to re-level lockpicking is to make use of the trainers (Ma'jhad with the Khajiit caravan, Vex in the Thieves Guild) and to break picks in Master locks. Every broken pick grants some exp, less if it's broken in a Novice lock, more if in a Master lock.
• Lockpicks can be pickpocketed from occasional NPCs, looted from occasional bandits, found in the Thief Caches if you have completed City Influence Quests, and bought in bulk from Khajiit caravans and thieves guild fences. The two most easily accessible Master locks for breaking picks in are the House of Clan Shatter-shield in Windhelm (make sure you store the key if you have it before trying to break picks in the lock) and the Master level chest in the Thieves Guild training area.
• Good luck.

PICKPOCKET

• Like Lockpicking, this skill can be somewhat hazardous to re-level due to the slowness with which NPCs restock their inventories. But it can be done, and easier than lockpicking.
• Delvin in the Thieves Guild offers infinite radiant quests, Fishing Jobs, that create an item to be pickpocketed. It is a good idea to save before generating a Fishing Job, to be sure that the radiant item assigned isn't too expensive for your skill level (e.g. he sends you to pickpocket a gold diamond necklace when your pickpocket skill is 19). Be sure too to save before turning it in if you have any other similar items, to avoid bugs where the ruby ring you pickpocketed isn't taken, but rather your very expensive and irreplaceable custom-enchanted one.
• Travel overland, meet interesting people, and pickpocket them. This is slower, because these NPCs rarely have high-exp items like jewelry, but as they're generated right then, they usually have something to take, as opposed to citizens who probably haven't reset inventories.
• Once you get to pickpocket 40, you can take the Poisoned perk. This allows you to harm NPCs by reverse-pickpocketing poisons onto them, and grants skill exp if successful. Dark Brotherhood Forever targets are particularly useful as they always have 50 health, and poisons that remove that much health are easy to craft or buy even with low skill.
• Trainers. After training a skill, pickpocket the cash back. Careful you have the Cutpurse perk and don't spend TOO much at once, so that you are capable of re-pickpocketing the gold.

SPEECH

• Buying and selling things increases skill exp. The bigger the transaction, the more EXP granted. Thus creating and enchanting items or selling expensive potions will level your speech skill more rapidly than selling only found loot.
• AVOID buying and selling in bulk. Because of a bug, the game will only award exp for 1 item of the type bought and sold, not all of them--e.g., buying or selling a stack of 7 Iron Armor only grants exp as if you had moved 1.
• Have Dragonborn installed. Complete the Black Book Untold Legends (found in Benkongerike) and choose the Black Market power. Summon the dremora merchant. If you are at a high level (which, if you're resetting Speech, you almost certainly are), his stock will consist entirely of Daedric weapons and armor, and there should be one enchanted Daedric weapon and one enchanted Daedric armor piece. Buy them. Because skill exp increases with the cash size of the transaction, and these are by far the most expensive purchaseable items in the game, they also afford the biggest skill gain per. This can of course also be done with any merchant who has an enchanted Daedric equipment piece in their stock, I simply advise using the dremora merchant because he is convenient (can be summoned ANYwhere) and reliable (will always have enchanted daedric gear).
• Every time you visit a merchant who stocks dwarven arrows, buy one and only one from them. Sooner or later, you will be visited by a courier who has a letter from Calcelmo in Markarth. Calcelmo wants to purchase your arrow. Once you get to Markarth, you can persuade Calcelmo to increase the gold he offers for this. This is one of the few repeatable easy persuades in the game, and can be cleared at 25 speech if you wear an amulet of Dibella and stop in the Markarth temple to get the Blessing of Dibella beforehand.

ALCHEMY

• The more expensive the potion, the greater the skill gain. Take alchemy perks, wear rings, necklaces, and gloves (circlets too if you don't have Aetherial Crown) of fortify alchemy, and make multi-effect potions whenever possible.
• Abuse of the fortify restoration loop will get you to high alchemy very fast. Cheap as hell though.