In Chinatown LA you will find some good Chinese restaurants and shops. The area is located around North Broadway.
Other ethnic districts are Little Tokyo, on Temple St with Alameda St; and El Pueblo de Los Angeles Historic Monument, a Mexican District with the oldest houses in LA.
If you are looking for a good market, you can go to the Grand Central Market, in 317 S Broadway, where you can buy fresh produce, meat and sea food.