Arts
The Los Angeles Central Public Library
Huge library rebuilt in the 1980s and '90s. Almost always has a public exhibition going.
Fashion Institute of Design & Merchandising
Gorgeous campus of FIDM and ongoing free exhibits make this a pleasant way to kill a couple of hours.
Music Center and Disney Hall
Impressive hall architecture complete with tours most days. The Dorothy Chandler Pavilion is open to the public Christmas Eve day with almost round the clock performances by amateur cultural arts groups. The Walt Disney Hall has daily tours, check website for schedules.
Museums
Geffen Contemporary
A branch of MOCA tucked away in Little Tokyo. Same opening hours and shared tickets as MOCA on Grand.
Japanese American National Museum
Covers the Japanese-American experience, with a special emphasis on the concentration camps of World War II.
Museum of Contemporary Art (MOCA)
The permanent collection is fairly interesting, but the changing exhibitions can be more hit or miss. The museum has no 'traditional' art, so bring an open mind. The gift shop free entrance is fun for at least 20 minutes of wonder and awe.
Old Plaza Firehouse
This was the original fire station for the City of Los Angeles. Built in 1884, it has been restored to its original condition. The knowledgeable docents offer a peek into Los Angeles in the 19th Century.
Cultural districts
Chinatown
Primarily centered around North Broadway; unlike Chinatowns in many other cities, it has a wide, main, busy street filled with small shops and restaurants. At about the middle point of N. Broadway in Chinatown is an open market much like those found in Hong Kong. Be sure to haggle!
Little Tokyo
Also known as J-Town, the Japanese district features restaurants, museums, and shops. It sits in the area between Temple and about 5th and Spring through Alameda.
Olvera Street
This is where LA was founded as El Pueblo de Los Angeles. You can take a tour of the city's oldest house to see what it looked like at that time. The plaza is mostly filled with Mexican trinket stands and Mexican restaurants.
The Bradbury Building
Built in 1893, the Bradbury Building is one of Southern California's most remarkable architectural achievements. Behind its modest exterior lies a magical light-filled Victorian court that rises 50 feet with open cage elevators, marble stairs and ornate iron railings. The building has been a set for many movies, including Blade Runner in 1982. Visitors without business in the building are allowed into the lobby and up to the first landing of the staircase.