When I start the emulation, everything's fine until I have to choose how I want to play. I try "Career" and "Free Ride" and stuff, but both A and B take me back to the "Press Start" screen. I can only go backwards from there.
Your input plugin is not configured correctly then.
Download this one and put it in Project64's /plugin/ folder, and in PJ64 go to Options --> Settings and choose it as your input plugin. Then configure it by going to options --> Configure Input Plugin.
No see this is the problem; up until a certain menu, it's perfectly fine. A takes me forward and B takes me backward. But once I get to that specific menu, A and B do the same thing.