The three times I've trekked to LA, I just got around with the LA Metro transit system. By far it'll be your most affordable option, though not necessarily your fastest moving. If you're willing to sacrifice some time to save a lot of cash, getting from point A to B is no more difficult than just punching where you are and where you want to go in Google Maps to get the transit links that will get you there.
I did rent a car once in one of my trips. I'm not sure what defines a cheap car rental in LA. I rented one for a day with a hundred-mile limit that came to at the end of the day, gas included about $75. If your intent is to travel in mostly urban areas instead of, say, out to Malibu, I'd go for public transit first.
As far as places to stay, while it may not be handy to anything aside from Hollywood Boulevard, I stayed at
this hostel on my last trip and was reasonably pleased, considering I've stayed at others in the city that were not nearly as nice for not much more in savings.
Other things to see... I'd make an afternoon for a trip to the Paley Center in Beverly Hills, and if Martin's working the counter, make sure you let him know you're there to watch game shows. He'll treat you like gold. If you've got enough time left before you go, I'd highly recommend a day at the UCLA Film and Television Archive. Watch disasters you'll see nowhere else like The Honeymoon Game and Duel in the Daytime, as well as gems including the third episode of The $10,000 Pyramid. Most of the material has to be requested several weeks in advance, which is why I say "if you have enough time left before you go."
Other must sees and do... I can only think of food off the top of my head. Don't skip In-N-Out Burger. Pink's Hot Dogs on North La Brea are the best hot dogs I've ever had. Oh, when you're done at TVC, head down Beverly Boulevard to El Coyote Mexican Cafe, rightly recommended by Roger Dobkowitz himself (no doubt in part because of their Margarita Mondays).
