You should get down Hermosa/Manhattan. The beaches are too cold/dreary/windy right now to enjoy, but if nothing else you can rent a bike or rollerblades and ride along the strand. I used to live there and it's still my favorite place in the LA Metro area. I actually prefer the South Bay to Huntington/Newport, which are way too West Coast Choppers OC for my taste. Not really fun to go out there unless you know people, but if you do know have friends in the area, pre-partying and hitting up the pier bars is A LOT of fun.
WeHo, if you're not 21 the only place you can get into is Rage. Micky's is my favorite bar to dance at, Here Lounge and The Abbey are also fun depending on the night.
If you're inclined to go out in Hollywood, Boulevard3 is the best club, but hard to get into.
Places to visit: hike Runyon Canyon (Hollywood) or if you're adventurous, Escondido Falls (Malibu). If you have time, just cruising up the Pacific Coast Highway through Malibu is a great time. From Malibu Canyon Road near Pepperdine you can loop through Canyon Country up to The Valley. There's horseback riding there if that's your thing. Or you could continue to Simi Valley to visit the Regan Presidential Library, which is really cool (and I'm not conservative).
On clear days, the Getty, the Getty Villa, and Griffith Park (Griffith Observatory) are must visits.
Downtown, you can walk around USC's campus which is really nice, the Staples Center/LA Live complex and then do some shopping on Olvera Street or in the Fashion District. Santa Momca's 3rd Street Promenade, Pasadena's Old Town, The Grove, and the WeHo Melrose district are also good places to shop/people watch.
Must eats: Roscoe's Chicken & Waffles, Crustacean (if you can afford it), Hucklberry in Santa Monica for brunch, Fogo de Chao, any of the Le Pain de Quodieten, The Little Next Door...there's a ton of good restaurants.