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.
Sarcasm is an allergic reaction to stupid people.
