Where to Stay
See my companion piece about Orange County on my Anaheim page. This will help you (hopefully!) decide if you want to make somewhere in LA or Orange County be your base while exploring SoCal.
Like OC, LA County can be divided into different areas. The main difference, besides the sales tax, which I discuss more fully on the Anaheim page (but basically, it's higher in LA County), is that LA County is much larger than Orange County. This just means more cities you have to get to know in order to decide where to stay and where to visit.
The South Bay is a region in the southwest corner of the county. Here, Palos Verdes Peninsula sticks out into the Pacific. The cities around the peninsula are Manhattan Beach, Redondo Beach, Torrance, Palos Verdes, Rolling Hills Estates, San Pedro, and Long Beach. For visiting both LA and Orange County attractions, if you want to just have one base for the entire trip and have that base be centrally located, your best bet would be one of the beach cities. The 405 fwy skims the eastern edge of Redondo & Manhattan Beaches and runs right through the middle of Long Beach. From the 405 you can get just about anywhere in either county with one or even no fwy changes. There's not much to see in these cities except the beaches - try Redondo early in the morning if you like dolphins - but, after a long day of sightseeing, a walk on the beach would be a great way to relax and all these cities can provide you with that. Long Beach also has a great place to kayak in Alamitos Bay and there's also the Aquarium of the Pacific.
Palos Verdes and Rolling Hills Estates meld seamlessly into each other. They are the 2 towns that make up the majority of the peninsula. This area is known for curvy roads, fantastic scenery, exclusive homes, and Wayfarer's Chapel (a glass chapel on the cliffs overlooking the ocean). It's a great place to visit but not really an option to stay. The roads are difficult and confusing and it's out of the way if you want to go anywhere else.
San Pedro is also on the peninsula. This is a small community where most of the cruise ships dock. There's a pretty lighthouse and the Korean Friendship Bell. This area has some hotels and motels but it can be a rough neighborhood and, again, its out of the way location doesn't make it ideal for a convenient homebase.
The Eastern area of the county really doesn't have much in terms of tourist attractions and making your base here would mean a long drive no matter where you wanted to go - beach, downtown LA or some parts of Orange County.
The northern beach cities would be Hermosa Beach, Santa Monica, Malibu. SM & Malibu are world known so there's much to see here but the prices are going to be higher. If you're not going to spend much time in Orange County (so you want to stay closer to the LA sights) but want a cheaper place than can be found in SM or Malibu, try Hermosa Beach. It's similar to Manhattan and Redondo but a little closer to LA.
The northeastern area includes Pasadena and South Pasadena. This is a lovely part of the county with large, beautiful homes, great shopping and restaurants, and terrific museums. It would also be a great place to set up your base. Just remember - a drive from here to the beach will take you a VERY long time. On the other hand, to get to Orange County you can hop on the 110 or 10 fwy to the 5 and you'll be in OC quickly.
Central areas - the central part of Los Angeles has a lot to see - the downtown architectural and historic sites, Dodger Stadium, Chinatown, Union Station, Olvera Street - but most of this area is not some place you'd want to find a hotel. If you want to be closer to the action and a little cheaper than Santa Monica, for example, and you don't need to stay right by the beach, consider Pasadena or Century City or the West L.A/Hollywood areas