Yes, there IS great live theatre in Los Angeles. As a once-upon-a-time theatre major and huge theatre fan, I was really disappointed in the theatre scene when I first moved to Los Angeles, but that was many years ago. In my admittedly opinionated opinion, things have changed drastically for the better over the last couple decades and I'm happy to say, legit theatre in Los Angeles no longer sucks. That is to say, there are still some miserably awful small theatre companies in LA, but there is also a lot of excellent live theatre in the Los Angeles area, both on the large professional stages, and at smaller 99 (or less) seat venues.
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Broadway in LA
You can see touring Broadway shows in several LA area theaters, which are the same shows you could see in Cleveland or Dallas. That's not a bad thing. Many of those performances are outstanding. And when you can pair a Broadway show with a visit to a gorgeous historic theater like the Pantages Theatre in Hollywood, it's bound to be a great night out in LA (even though they set their sound system too loud).
Note: Seeing a Broadway show in LA is not quite the intimate experience you get on Broadway because these theatres that host the touring companies are significantly larger than the biggest Broadway theatre. Balcony and mezzanine seats that might still have a good view on Broadway are much farther from the stage in the big LA theatres.
Los Angeles Companies Produce Some Great Theatre
If you already saw those shows when they passed through your town (we're usually at the end of the national tours) you can get a sense of home-grown LA theatre produced by local companies putting on outstanding productions of classic plays and musicals as well as new works making their world premiere.
TV and Movie Stars on Stage
In a city with so many TV and movie actors, it's no surprise that you can often find well-known celebrities in LA stage productions (the fact that some big names will do legit theatre in NY, but not LA is another story). Sometimes this is wonderful, and you get to see a totally different dimension from a talented performer. On other occasions, the big name is an attempt to draw in a bigger audience, but the casting is not a good match or the skill set just doesn't transfer and the overall theatre experience is disappointing. In any case, I applaud those screen actors for putting themselves out there in front of a live audience with no second take.
As a tourist, the place to check first, for theatre quality, accessibility and potential celebrity casting, would be the Center Theatre Group, which stages both popular and new works at the Ahmanson Theatre and the Mark Taper Forum at the Los Angele Music Center, as well as the Kirk Douglass Theatre in Culver City. The Music Center is an attraction in itself, and can be a nice focal point for a weekend getaway to Downtown LA.
The Geffen Playhouse, just off campus near UCLA is another theatre that I admire for their willingness to take risks with new work and bring more intellectual works to town.
They also have their fair share of TV and movie actors gracing their stages.
Los Angeles Theatre Gems in Small Spaces
There are so many small theatre companies doing excellent work in the North Hollywood Theatre District, in Santa Monica, Hollywood, Long Beach and everywhere in between. Any given weekend, there are literally many dozens of plays and musicals being performed in greater Los Angeles.
The spaces range from tiny store-fronts with 25 folding chairs to small 99-seat theatres and a handful of medium-sized theatres with 2-300 seats. Some of these are in their own space, like the Santa Monica Playhouse, Theatre of Note in Hollywood, El Portal Theatre in North Hollywood, International City Theatre at the Center Theatre in Long Beach, the Long Beach Playhouse, and A Noise Within in the Masonic Temple building in Glendale.
Other theatre companies rent space as needed, so they may share space or move from location to location.
So when you're looking at reviews of plays, it's more important to pay attention to the theatre company than the physical theater, because the next time you see a listing for a show at that theater, it might be put on by a different company.
Live Theatre is a Bargain in LA
The average price of a movie ticket in LA is $10-$16. Outside of the largest venues and Broadway touring shows, which can be quite expensive (but still less than comparable tickets on Broadway), the average price of a live theatre ticket in LA is $20-$30, with many as low as $10 for a full price ticket. Given that the supply of theatre seats far exceeds demand, there are always half price tickets available to a couple dozen shows (see LA Discounts for info), bringing the prices down even further. I know it's sacrilegious of me to say in a movie town, but if I can see a live performance for the price of a movie, I'll always choose to spend my time at a live performance.
Even though those full price tickets for big musicals can be pricey, when you look at how many bodies on and off stage make that production happen, your $120 ticket is split many ways to keep a lot of people employed. So I encourage you to pay full price when you can, and enjoy being a true patron of the arts.
Theatre is a Weekend Event in Los Angeles
Unlike New York, where many shows run 6 days a week, in LA, live theatre is largely only available on weekends. A weekend in this case is defined as any combination of 2 or more days between Thursday and Sunday. If you want to see a show on a weeknight (Mondays are dark), the Pantages Theatre, the Center Theatre Group or the The Geffen Playhouse are pretty much your only options.
Los Angeles Theatre Discount Tickets
There are several outlets that offer discounted tickets to Los Angeles Theatre productions. There is some overlap, but some tickets are only listed at one outlet, and they have different service charges, so check all of them before buying. You can also sign up for their email alerts to be notified when new shows are posted.