Depending on what your class is, pick these skills
For a thief like class, pick Alchemy, Light Armor, Sneak and Security. To level Alchemy quickly, obtain an Apple and some cheese. Go to the Mystic Emporium and buy 2+ of the same scroll. Use the scroll dupe glitch on the Apple and Cheese, and dupe many. You probably need 600+ of each ingredient to level your Alchemy to 100. Leveling your alchemy to 100 will get you to level 7.
To level Light armor the quickest, simply go to Arvena Thelas's house, go into the basemant, and unequippe ANY weapons and turn the difficulty slider to very hard. Punch each rat once, and then turn the difficulty slider to very easy. Let the rats beat on you until your Light Armor skill is 100. This won't take long, and be sure to have light armor equipped, and you need to occasionally drink a heal potion or cast a restoration spell.
To level Sneak quickly, go to the Inn of Ill Omen, and go to the corner where the trap door is. Press the auto-move button (Q is default) and go into sneak mode. Make sure the eye in sneak mode is not bright, and that there are people around. It doesn't take to terribly long, but it still takes a little while. If you do this, go watch TV for awhile. You will NOT level your skills if you alt-tab out.
To level Security easy, use the scroll dupe glitch and dupe a few hundred lock picks. Find an Altar of Spellmaking (If you do not want to join the Mages guild, download this mod: http://www.tesnexus.com/downloads/file.php?id=15116) and create a Drain Security 100 on self spell. Go find a lock door, cast the spell and quickly go into lock picking mode. Press auto-attempt over and over until you run out of lock picks. If this happens, dupe more. You will level security very quickly.
Doing these steps will get you to level 20+ very quickly. Once you get to level 20+, you can go to the Roxy Inn and kill the NPCs inside to get Daedric and Glass armor.
For Warrior Class
To level Heavy armor the quickest, simply do the same thing as I told you to do with Light armor, just wear heavy armor this time.
To level Blade or Blunt quickly, go to Peryite's Shrine. Equip a sword or mace, and bring many repair hammers with you. Beat on the followers at the shrine. They cannot die, and will stand there while you beat on them. Note: This takes a long time to level your Blade or Blunt skills to 100. You need a lot of patience. Also, use with caution. Beating on the followers can possibly cause an unstable game and crashes.
There are a few less skills to level with a Warrior class, and you'll probably won't get as many levels as fast as you would with a Thief or Mage class.
Leveling a mage fast, it's simple. Pick the Mage class, Apprentice Birth Sign, and get a few mana potions. Pick a spell in the class you want to level, and find an object to hold down the cast button with. After a few levels, you probably won't need any mana potions. And, if you don't mind cheating a bit, bring up your console if you're on a PC and type in tgm. This way your mana won't deplete and you can cast until the cows come home.
just some tips I've picked up through 3 years of playing that I thought I would share. Hope it's useful!