whatever you do, don't miss out on the great ethnic foods in the city/region. mexican, thai, korean, armenian, persian, salvadoran, guatemalan, and japanese are just a few of the notable options in the city proper. really outstanding chinese food (cantonese/hong kong-style, northern mainland, and taiwanese) in the suburban towns of the san gabriel valley and awesome vietnamese food in orange county. indian food in the suburban town of artesia. more armenian in glendale (that's the armenian capital of southern california) and more japanese in torrance/gardena. i could go on and on. many of these ethnic cuisines are better in the LA area than anywhere else in the country.
if there's one ethnic cuisine you shouldn't miss, though, it's mexican. there are so many great restaurants, taquerias (taco shacks), taco trucks, etc. i'd especially try to get some fish tacos, since they're not available in most parts of the country outside of LA/orange county/san diego. my favorite fish taco shacks are tacos baja ensenada on whittier blvd in east la and baja fish on s. gaffey st in the san pedro neighborhood of la (closer to long beach), which is also known for its fish burritos.
after that, thai food is probably the second "must eat" cuisine in LA. thai town is conveniently located in the eastern part of hollywood, and has some of the best thai food in the country. IMO the gap between the thai food in la vs. any other city/region in the u.s. is huge.
and at least for me, mashti malone's in hollywood is a must-visit for exotic persian ice cream.
as for non-ethnic foods, you should not miss in-n-out burgers. not only are they great, but they're classic southern californian fast food. you can skip the other so cal chains like carl's jr., jack in the box, el pollo loco, or del taco, though.
other things that are must-dos:
-it was mentioned already, but driving north up the PCH from santa monica up through malibu is a classic southern california experience, preferably in a convertible. the beach on your left and the mountains on your right make for awesome scenery. you can drive up some of the mountains and take great pictures of the ocean, and if you hit the right spot, you can also take pictures of the city of LA in the other direction.
-mulholland drive is another great road to drive on for amazing views. once again, bring your camera. you can take mulholland from the coast all the way to hollywood if you have time, or you can just drive along a portion of the road.
-this is obvious, but you can't miss the beaches. there are numerous beaches in malibu if you're heading up that way; otherwise, the south bay beach cities (manhattan beach, hermosa beach, redondo beach) are great, too. santa monica or venice are good if you want to walk
along the beach/boardwalk and people watch, but do not want to actually go
into the water or lie on the sand. lots of stuff to see and do and very crowded with locals and tourists alike, but the beaches themselves are among the dirtiest in southern california. people go to sm and venice because they want to be near the beach while hanging out on the santa monica pier or 3rd street promenade.
outside of LA county, there are great beaches in orange county. pretty clean for the most part, too.
-continuing the theme of beaches and food, here's a link for some eateries/cafes along the beach with great views:
Elina-shatkin's list of Best L.A. Beachside Restaurants
-if you're a hipster or into indie music and live shows, go to the silver lake, los feliz, and echo park neighborhoods.
-you have to drive down the sunset strip at least once. again, preferably in a convertible.
-personally, i like the ucla and usc campuses. if you have time, it might be worth checking one of them out.
-old town pasadena is a really cool place to eat, hang out, and people watch.
-the dodgers' season is now over, but for future reference, catching a ballgame at dodger stadium is a must, even if you're not into baseball. it's such a beautiful, classic stadium that's been wonderfully maintained over the years. seriously, it feels like a relic from LA's golden years in the 1960s.
if you're ever down in orange county, angel stadium is also nice and is in the same city (anaheim) as disneyland. and anaheim's pretty close to the orange county beaches and vietnamese food, in case you're looking to spend an entire day or two down there.
-if you're into basketball, a lakers game might be a must-do as well, but even though the staples center is really nice, i don't feel that it's as worthy of a visit as a baseball game. i dunno, i could be off on that. but to me, there's something really special about going to a baseball stadium when visiting a city - each of them are unique in their own way, whereas basketball/hockey arenas and football stadiums across the country are not nearly as distinct from one another.
-as cheesy as it sounds, you might want to check out the chinese mann theater in hollywood. kind of like times square in nyc - overrun with tourists, but many people will say you need to see it at least once. at the very least, drive by and take a quick photo or something.
right outside of the theater, there are tour buses that, for $20 or $30 (not sure), will drive you around the hollywood hills, beverly hills, bel-air, holmby hills, and other super wealthy neighborhoods to gawk at various celebrities' homes from outside their gates. i'm ashamed to admit that the first time i visited LA, i did this with my friend (an LA native), who'd always wanted an excuse to see the stars' homes but could never justify doing it by herself. although i felt sheepish during the two hours, i have to admit, it was sort of fun in a trashy way. and not terribly expensive or time-consuming, either.
-even if you're not into shopping, you should walk down rodeo drive at least once, just to say you've done it. again, this is something that the locals might not do on a daily basis (who can afford to shop there everyday other than the super rich?), but as a first-time visitor, it's a way to get a fleeting sense of the stereotypical la glitz and glamour.
as you can tell, most of my suggestions revolve around eating, cruising around, hanging out at the beaches, or checking out scenery. i'm sure others can give you plenty of other suggestions re: nightlife, culture, etc. have fun.