Fastest ways to level multiple skills in Skyrim

This is a pretty broad/general question, so this is going to be a pretty broad/general answer. Remember that you can activate the relevant Guardian Stone to get a skill learning boost. Many of the skill pages on the UESP contain "tricky" or "glitchy" ways of gaining skill ranks very quickly. Also, take a look at the mod workshop if you're on PC, since there are likely things there that will make this process (or the game) easier, if you want to waste less time maxing everything out. Console commands would also make this easier, or you could exploit a glitch. I'm guessing since you asked this question, you're wanting to level legitimately, though.

Magic Skills: Alteration/Conjuration/Destruction/Illusion/Restoration

Skill gain is related to the power of the spell as it effects valid targets. For instance, with restoration, the amount of health healed, or with destruction, the amount of damage done. For illusion, rally friendly townspeople. Conjuration is easily leveled by getting into combat against something that can't hit you, and then repeatedly casting bound weapon/armor spells.

Combat Skills: Archery/Block/Heavy Armor/One-Handed/Two-Handed/Light Armor

Skill gain is related to damage done or absorbed. Do a ton of damage to something that's invincible, being healed, or has a ton of health. Giants, for instance, or essential people from quests who don't care if you pummel them. Shadowmere is a decent punching bag as well. For armor/block, absorb as much damage as you can without dying, as fast as you can. Potions and spells can offset this damage.

Crafting Skills: Smithing/Enchanting/Speech/Alchemy

Skill gain is related to the value of the item involved. The more expensive the transaction or item created, the more the skill gains. Make expensive items, or make large batches of cheap items. Better yet, make large batches of expensive items :) Fast travel in a circuit to various raw good merchants (mages for soul gems, blacksmiths for smithing components, alchemy vendors for ingredients, etc), buy up their stock, craft, and sell whatever you can in town before taking off to the next town.

Stealth Skills: Sneak/Lockpick/Pickpocket

Sneak skill gains are related to use of the skill, and major gains from backstabbing. Sneak attack sleeping enemies (Draugr are easy to find and frequently napping in their crypts).

With lockpicking, breaking picks gives you a small boost, so you can throw picks at the problem if you're flush with cash. The biggest gains come from opening high-level locks.

Pickpocket gains are related to the value of the item, whereas difficulty grows as the value goes up and as the weight goes up. Steal many light, but relatively expensive, items. Quicksave often, as this is the most likely skill to get you into hot water.

Strategy

I'd probably focus on the crafting skills first. Join the Mages' Guild and the Thieves' Guild so that you have access to the mage vendors, the radiant thief quests, and the guild fence. Run jobs when you're low on money, and while you're moving around the map, steal everything that isn't nailed down in every alchemy and blacksmithing shop. Don't forget to rob the court wizard blind while you're at it. If you break in to shops and run numbers job/heist job quests, you'll have plenty of opportunity to level stealth skills.

Once you've got a fair bit of gold and the skills to make some decent gear, go run the assassin guild questline to get Shadowmere and the guild's torture room, which you can only buy once you've gotten to the end. You can abuse the torture victims to level your offensive skills.

Now that you've got awesome armor and fearsome offensive skills, go find a giant and pick a fight. Drink potions of health restore/regeneration to avoid death, and watch your skill levels rise.

The quickest way is to use the Oghma Infinium exploit. The first steps toward getting this book are to get to level 15, and then start the "Discerning the Transmundane" quest by going North of Winterhold to Septimus Signus's Outpost.

Basically, you get the book and take it back to your house without reading it. Activate the bookshelf and read the book. Then go back 1 (one) screen, and put the book on the shelf. Then take the book off the shelf without reading it. Then activate the bookshelf again and read the book. Then rinse and repeat. You'll be able to get all of your skills up to 100 in about 15-20 minutes or so.