Does killing enemies give XP in Skyrim?

No. Only skill level-ups raise your character level. Kills help only because you use skills while fighting.

Which skill is the cheapest to max?

By UESP data, Pickpocket, Smithing and Lockpicking need the least total skill XP to go from 15 to 100.

Can I change my Health, Magicka or Stamina choice?

Not in normal play. The choice is permanent unless you use PC developer console commands.

Is there a level cap?

Before patch 1.9 the cap was 81. Legendary skills now allow you to keep going, according to UESP.

Skyrim Leveling Guide: Which Skills to Level First

Skyrim has no classes and no XP from kills. You get stronger by using skills, and your overall level follows. This guide explains the rules in plain words, then shows which skills level cheaply and which ones are worth your early perk points. The numbers come from the Unofficial Elder Scrolls Pages (UESP), a fan wiki that tracks the game's data.

Key takeaways

Character XP comes only from skill level-ups. Nothing else raises your level.
Every character level gives one perk point and +10 to Health, Magicka or Stamina. The attribute choice is permanent.
Smithing, Lockpicking and Pickpocket need the least XP to reach 100. Most combat and magic skills need the most.
Enemies and loot are leveled, so raising crafting skills only can make fights harder than your gear can handle.

How the two kinds of leveling work

Each of the 18 skills has its own bar. You fill it by doing that skill: swinging a sword raises One-Handed, casting raises your magic school, crafting a dagger raises Smithing. When a skill goes up, your character earns XP equal to that new skill level. The XP needed for the next character level is (level + 3) x 25, so early levels come fast and later ones slow down.

These screenshots may contain violence.

Images are provided by Steam and belong to their publishers.

Cheap skills versus expensive skills

UESP lists the total skill XP needed to go from 15 to 100. Here are rounded figures from that page:

Skill group	XP from 15 to 100 (about)
Pickpocket	87,000
Smithing, Lockpicking	91,600 each
Sneak	142,000
Enchanting	279,000
Alchemy	429,000
Most others (weapons, armor, Block, Speech, magic schools)	529,000

Cheap does not always mean fast to use. Pickpocket and Lockpicking depend on finding things to steal or open, and Sneak only grows while you are unseen.

Which skills to level first

Your main combat skill. One-Handed, Two-Handed, Archery or a magic school. You use it all game, so its perks pay back right away.
Smithing. Better gear keeps you alive, and it is cheap to raise. Spend early perk points here only if you will really craft.
Alchemy or Enchanting. Both can boost your power, but they are slow to pay off. Pick one, not both, at first.
Sneak or Block. They give safe options in tough fights.
Speech. Handy for prices and a few quests, but it can wait.

Health, Magicka or Stamina?

Each level adds 10 points to one bar. A common rule: warriors favor Health or Stamina, mages favor Magicka, but many players put most points into Health because it keeps them alive. You cannot undo the choice in normal play.

Do not out-level your gear

UESP notes that most enemies and loot are leveled. If you push a crafting skill very high and little else, your level rises while your fighting skills lag, and enemies keep up with your level. Spread your skills, and keep your weapon and armor skills growing.