Playing in musical theatre is a great way to get real live experience in a wide variety of playing styles from rock to funk to jazz and just about any other style. So how do you start out? If you are very lucky, a very good reader and player, and know someone, you might be able jump right into a paid gig. Don't count on it though.
Scott Devine has some good suggestions on how to get gigs by getting to know the in demand local players. His video is worth a watch.
In some places you can sit in on a professional pit if you find who the player is and get permission from the musical director (the one that runs the singing and playing, not the one that tells the actors what to do). In practice, this may not be practical because of space constraints. Theatre pit space tends to be at a premium.
If you are a good player and a good reader, maybe the best way to get experience is to check out the local professional, regional and community theatres. Pro and regional would be paying gigs, and community theatres may be paying, but likely not much. If you have the time to devote, try it out for experience. Google is a great way to find theatres.
I have said "good reader" a couple of times here. While most shows have cast albums available, there is usual a great deal more to play live than is captured on the album. This is true especially where the show has a lot of dance numbers. The album may have 40 bars of dance break, but the actual show has 240 bars. Point being that if you can't read music, you are going to have a hard time. It is very difficult to learn these by ear, and you will be expected to play what is on the page. The take-away is to learn to read music, and practice sight reading. A good book for this with a lot of different styles is Readings in Contemporary Electric Bass from Berklee Press.
Plan on being auditioned by musical directors the first couple of times out. That will most likely consist of being shown the bass part and asked to play on sight. (See the last paragraph). What they are looking for is accuracy and rhythm, not necessarily up to tempo. It may be a song you have never heard of, so unless the style is obvious, you may not that on the read. The parts tend to be in 2 to 4 bar phrases, depending on the meter (Songs in 2 are likely to be 4; ones in 4 are likely to be 2) particularly in dancey numbers, as dancers like tocount phrases of 8 beats. Take your time and concentrate on rhythm and accuracy. Playing too slowly will not get you the gig, most likely. Think 60 to 100 bpm depending on what the mood of the song seems to be.
Once you get a gig you need to honor the commitment, whether it is paying or not. As a part of the rhythm section, the actors and other musicians need you to be there. Community (amateur) theatres may or may not be forgiving of other commitments, but once the opening is about two weeks away, you will be expected to be there. An average show bass book will have anywhere from 20 to 40 songs, and is a bit daunting at first. You will need to have worked through the book before you go to a rehearsal.
Lastly, an MD will appreciate an honest question about what he or she expects. It is better to ask than to guess and be wrong. No one will fault you, no matter how experienced you are.