One cannot visit Downtown Los Angeles without going to Olvera Street. Whether tourist attractions are your thing or not, the feel of this place is definitely something to check out to realize how close California is to the Mexican border.
But before checking out Olvera street, which to me could also be called Little Mexico, there is two building close by that are worth seeing. The first one is the Los Angeles Union Station on North Alameda Street.
The second great piece of architecture is the U.S. Post Office - Los Angeles Terminal Annex also on N.Alameda Street.
The particularity of Olvera Street is that it's the oldest part of Los Angeles!
The size of trees can testify of how old this part of L.A is.
Of course, the one attraction on Olvera street is its market full of kitsch and traditional Mexican items. The Lucha Libre (wrestling) masks were for instance in almost every booth which I find both funny and scary.