Does my race decide my build?

No. Each race has skill bonuses and a racial power, but you can level any skill.

How many skills are there?

Eighteen, split evenly among combat, magic and stealth.

Yes. Skills level through use, so you can mix warrior, mage and stealth.

Skyrim Races and Builds: Warrior, Mage and Stealth Basics

The big idea of Skyrim is freedom. There are no classes, only choices. This guide uses Wikipedia's description of the game to explain how a build grows from your actions.

Key takeaways

There are ten races, and each has skill bonuses plus a racial power.
18 skills are split evenly among combat, magic and stealth.
Skills level by use, and each level-up gives a perk point.

The Elder Scrolls V: Skyrim – Official Trailer · YouTube

The ten races

Humans: Nord, Imperial, Breton and Redguard. Elves: High Elf, Wood Elf and Dark Elf. Then the Orc, and the beast races Khajiit and Argonian. Each comes with a few starting skill bonuses and a special power, such as the Nord's Battle Cry. We do not list every bonus here because we could not verify them all, so check the in-game race screen.

Three simple build ideas

Warrior: Use melee weapons and armor. Combat skills level while you fight, and perk points go into the skills you use.
Mage: Use spells. Magic skills grow as you cast, and your Magicka bar matters most.
Stealth: Sneak, sneak attack and pick locks. Stealth skills grow as you avoid being seen.

How perks work

When your character levels up, you can raise Health, Magicka or Stamina and spend one perk point. Perk points unlock bonuses in the skill trees. Spend them where you spend your time.