If you live outside of NYC, then you will want to work at Equity theatres in your city (or nearby major city) as a local, non-Equity actor--often in the chorus. You earn points per week of rehearsal and performance. After 50 points, you can join Equity. In the meantime, you are growing on the professional stage and earning the experiences to fill out your resume.
You can also be offered an role that happens to be on an Equity contract. Unless you are in a right to work state, you will have to join Equity to take the role. Joining outside of NYC is a tricky question as there isn't as much work for Equity actors.
I would join before coming to NYC if you can. Otherwise, you can do non-Equity auditions and try to be seen at Equity auditions here.
It's really all about timing. And, yes you have to join Equity to be on Broadway, Off-Broadway or an Equity National Tour. Lead roles in regional theatre are also Equity.
AEA is the union of stage actors and stage managers. For film, you would join Screen Actors Guild. There are also unions for radio/tv and opera, etc.
You can find much more info at the AEA web site. And, feel free to ask any other questions---this really is a lot to try to cover in one post. :)