Pushing the button will lock the other person out.
If it is correct it will add 1 to the persons score and go to the next question. If it is wrong it will just go to the next question. I am not sure how I will show the score.
How should I store and check the answers?
I'm planning on using servos as pointers to represent the score.
I don't have an LCD already,
Right now the best thing you can do is find examples, play with them, learn from them and make your own code. If you have any issues, let us know, but to get help, you must present a code with comments and put some effort into it.
Do you have any idea on how the MP3 Player will interact with the arduino? Have you worked that out already?Otherwise you'll need to hack it to get at the information somehow, so you can sync the Arduino with the MP3 Player.