Talent Points
Talent points are simple and easy to use. They help make you better a specific skill set of spells (such as your frost spells, or fire spells). Some of the time they are effective across the board (arcane has a lot of effective across the board talents).
Simply put, invest and focus in one tree while you're leveling. I prefered focusing on Frost for the increased survivability, and later on the AoE grinding was a blast.
Talents will help make your character stronger, however be careful in what talents you choose, it costs gold to reset your talent points and this can become costly.
A lot of people prefer speccing fire until 40, and then switching to frost. As I've already said I preferred to spec frost all the way to where I am now, but it's up to you.
Spells
If you've specced fire, eventually Pyroblast will become your solo opener. You'll probably move to Fireball Fireball Fireblast. In groups, Scorch + Ignite should be your main damage.
If you've specced frost, Frostbolt will be your damage dealer. You'll be spamming Frostbolt, slowing the mob (enemy), and stepping back as it gets close. This is called kiting. Eventually you'll freeze the mob in place, and Frostbolt+Fireblast it for huge crits (when you get shatter). In groups you simply spam Frostbolt, and sometimes use Frostnova and Fireblast.
Gear
Pre 40 focus on +int, +stam, +spirit. After 40 focus on +spell damage, +crit rating, and +damage.
Good Group Etiquette
Be sure to always offer food/water when you group, and be sure to 'buff' up your group with Arcane Intellect and when the situation calls for it Dampen Magic/Amplify Magic.
***
If you have any other questions, or if this was complicated feel free to ask. Hopefully you have fun, and welcome to World of Warcraft.