If you are visiting LA, you really need to rent a car. Distances in LA are huge--and this is what most people coming from overseas don't understand. To give you an idea, the area of LA alone (not counting its over 100 suburbs) covers an area that is approximately 80 kilometers long by 50 kilometers wide. It makes London (which is also quite large) look like a postage stamp by comparison.
There are four major airports just in the LA area alone --since people don't want to have to fly into the main airport on the west side of town and then have to travel to their suburb home over 65 kilometers away on the east or south side of town. The downtown portion of LA is about 30 kilometers from the airport and a taxi will cost you at least $60 one-way to get there. On the other hand, if you rent a car, it will cost you maybe $30 per day. Also, the way the bus system works in the LA area is stupid, since every city has its own system--and they don't work together well--and unlike other metro areas in the US, the LA area has no master rapid transit organization (meaning no one to get the different cities to work together with their schedules). Thus, the only good buses to go long distances are all run by private companies.
But you really shouldn't plan to stay downtown unless you this is a business trip. The better areas to stay are near the beaches (even Long Beach is okay if you stay in the "Belmont Shore" area of Long Beach--which is very nice)--although the preference would be for Santa Monica, Marina del Rey, Manhattan Beach, Huntington Beach, Newport Beach (my favorite--but a long way from the airport) or near one of the major amusement parks (either in Anaheim or in Buena Park--near Disneyland or Knotts Berry Farm).
And the best shopping is in the Fashion District in LA--I posted on that on another thread on here--so just search through ones on LA and you'll find it easily.
