You need to make contact with the sports editor or assistant sports editor, or assignment editor (or someone else who can put you in touch with the sports editor or other sports-desk personnel) at the AP city bureau closest to where you are, and then go from there. If you have no idea who anyone there is, contact the bureau directly and just ask who's in charge in sports (or news, if you're interested in that) and a phone number and/or email address to reach them if possible.
As for assignments, usually you will be called upon to write short, straightforward, round-up-type items ranging anywhere from a couple of graphs to separate gamers of maybe 8-14 inches, sometimes with a quote or two, but usually not. Most often, the assignments will be gamers, especially at first, although it is possible once you've worked for them for a while to be asked to do/or for you to suggest features or short news stories.
Often, if you are still in college or otherwise well-connected to a university, you may cover your school's teams (usually basketball, but also, sometimes, baseball, volleyball and other niche sports, particularly if they go deep into postseason). Or, you could, perhaps, also cover other schools if there are multiple colleges nearby, and/or if AP doesn't happen to want you covering your actual school. You should expect to write whatever they ask for, plus provide a full box score/stats (which you can usually have provided to you from event SID personnel after the games).
If there are freelancers in your area that you know (or could meet/get in touch with) you might see if they have ever done any work for AP, and if so, ask if they have any contact names/phone numbers for you. Or, you might check with some of the journalism department professors or internship coordinators at your university. Sometimes, they have contacts with local-area industry professionals and can put you in touch with someone.
There is an annual-edition book of newspaper/media publishers, editors and contacts that's been put out for years that you can order and buy with contacts of every paper and most media outlets in the country, and I'd suggest you do that, too. You might not use it often, but it can be handy to have for starting points.