Will somebody please explain the tactics trainer to me? What is the goal of the problems, checkmate? best possible move? And once I try a problem, how do I go to the next problem? Please help me understand this.

Yes, the tactics trainer is a series of positions where you are trying to find the best move, sometimes leading to checkmate or winning material or defending against opponent's threats. After you solve a problem you should see a big green button which says "Next Problem" which you can click.

OK, thanks a lot Patzer!