I stayed in the Hilton near Little Tokyo in the downtown area, it was a beautiful hotel despite it being near Skid Row.
For sight seeing I would recommend having a meal at the Chateau Marmont (a bit expensive but you could spot a celeb and the hotel is BEAUTIFUL), the Santa Monica Pier, Amoeba Records, The Broad (a new art museum that is free to enter), LACMA, explore Hollywood, Universal Studios.