Can you chat on xbox live while playing two separate games?

I just made a live account yesterday and I want to chat with my friend on there while I play GTA. She's going to be playing Minecraft. Can we do that, or does it have to be the same game?

You can chat with her if you are playing different games. Go to her profile and press "invite to party".
You can invite her to a private chat, which is only 2 people chatting (xbox live silver members can only chat). Just go to her profile, and near the bottom there's an option that says "invite to private chat". If you are playing the same multiplayer games, most games have a built in voice chat with other players in the same lobby as you.

You can do it in party chat but you both need to have gold subscriptions

All you have to do is start a party and invite the other person. Or get invited by your friend. It does not matter what either of you are doing as long as you are connected to Xbox Live.