You didn't say what city you're in.
In New York and LA, Naked Angels (which only has an actual theater in NY) runs Tuesday night readings where writers select actors to read their material (the moderators schedule the writers). This just ended in LA for the summer, and I'd guess the same is true in NY, but there are other groups like that, and groups that do play readings.
First Stage in LA for instance organizes staged readings of plays.
There are also places you might know from stand-up where people do spoken word. One in LA was at the Unurban Coffeehouse. It might be taking a break right now, but you could check there or at similar venues in NYC.
With the Internet these days, you have the wonderful opportunity of putting stuff on video and uploading it to Youtube. Develop some characters and let them talk on YouTube, or get some friends together and write short routines you can film.
Read Backstage, Actor's Access and other sources for casting calls. The very first time I got booked as an actor (paid!) I showed up without even a headshot, answering a Backstage ad in NYC. There's always something where your type is more important than your experience.
Classes of course are a good place to start, as is volunteering for theater companies (to take tickets, work concessions, etc.) Lots of other possibilities have been discussed on threads for beginning actors.