Depending where you are in L.A. there are tons of beaches and forests. Angeles crest is great in the summer time, but try to go the campsites at higher elevations (6000+). Chilao, Buckhorn, Vetter are all great, you'll feel like you're in the sierras, but without the drive. There are a few nice ones that aren't as high up in the mountains, but they tend to be more crowded with less trees. For beach camping I think you'd probably have to go either north torwards Santa Barbara, or south OC. acourvil mentioned a really nice one, but as he said it's reservations are really hard to get at the nice campsites, which is why I haven't been beach camping for a long time. There's also camping near lake casitas and castake if you're into fishing. Sequoia National park is a little further drive, but well worth it. And even though you didn't mention deserts, desert camping can be a lot of fun. It's not always a bare wasteland like you'd imagine.
If this is your first time camping, I recommend going with someone else who at least has some experience. Not only will they have a lot of the equipment, but it will just be a lot easier for you. Are you going by yourself, or with a whole family? The first impression will basically set your attitude torwards camping trips in the future. If you have a miserable first time, you're not going to be too excited to go again, so it's kinda important that you plan carefully. I take a lot of non-campers to Yosemite for their first time, simply because it's not really camping, it's kinda like staying at an outdoor disneyland. It has all the comforts of a hotel, while still giving you the feel of camping.
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