If you're into history Olvera Street is a must for Los Angeles. It is the birthplace of Los Angeles and known as El Pueblo Historical Monument.
I will spare you the history lesson, but you can tour the oldest existing house in LA built is 1818, Avila Adobe House. There are about 8 more historic buildings however they were under construction during my visit. In addition to the buildings there is a marketplace and countless restaurants to get food and drinks! Also on weekends there is usually music in the Mexican Style Plaza area.
The festivities at Olvera Street are free and it's open from 10am to 7pm.