Just made all my travel arrangements for Los Angeles. I'm super excited to be going. I've always meant to take a trip to LA and San Fran, but the opportunity never really presented itself till now. I can't wait. For those of you who know LA, I'm gonna try to stick it out without renting a car, is this a completely crazy idea?
"What a mystery this world. One day you love them and the next day you want to kill them a thousand times over." The Masked Bandit in THE FALL
I have never been to LA, but San Fran is extraordinary. I visited for the first time this year, and fell absolutely wildly in love. I hope you like it.
"For those of you who know LA, I'm gonna try to stick it out without renting a car, is this a completely crazy idea?"
Their public transit system is clunky and rudimentary, and it isn't a pedestrian-friendly town, but taxis are plentiful - though they usually have to be called, as there aren't tons of taxi stands and I don't think you can hail them. Try to get train maps and bus routes in advance if you can.
The simplest answer to your question is no, you can't get around LA without a car. Yes, there is public transportation, but it is (compared to the overall size of LA) extremely limited and you will spend most of your day trying to get from place to place and there will be a lot of places you can't get to at all. For example, to get from downtown to the beach you could take the Wilshire bus, but it would take about three hours. Each way. As a tourist, it's just not worth it.
There are taxis (although I would not say they're prentiful), but the only places I've ever seen a taxi stand outside of the airport is at a few of the biggest hotels downtown. They just don't exist anywhere else in Los Angeles. As ghostlight pointed out, they can be called, but you will spend an obscene amount of money on them because distances are so great. For example, if you stay downtown and want to go to the beach, a taxi will cost well upwards of $30 each way.
Just rent a car. It's the only practical way to get around.
Seconding Humbug. Unless you're planning on staying just a few days in a very small area (and thus missing a bunch of the tourist sites), you really do need a car in L.A.
I've lived in LA almost 50 years - if you have any questions about where things are, and how to get around, etc., feel free to ask. I love to give advice.