One warning…downtown LA rolls up the sidewalks pretty early. But Little Tokyo stays open later…lots of great Udon places. On the roof of the New Otani Hotel there is a beautiful little surprise …a Japanese garden. the restaurant there, 1000 Cranes, is too expensive and not worth it. And I once ate in the restaurant in the lobby there and my date broke his tooth on a shell in the seafood pasta. They acted like we were criminals trying to fleece them for a free meal. So don’t eat there, but take the elevator to the roof and have a nice walk.
The best Chinese restaurant in Chinatown has the suggestive name of Won Kok. It’s off Alameda. But I often just roam Chinatown getting Dim Sum at various delis.
You will also be right by Olvera Street. Touristy, but amazing Mexican food and some cool shops and architecture.
For a "fancy" meal, Cafe Pinot and Cicada are great. For an expensive, but once in a lifetime steak..
This place is supposed to be great, and union Station has interesting architecture:
My dad used to take us here when we were little…seems like you are in the middle of nowhere by the UPS center, but very cool:
For a truly off-the-beaten-track mom and pop place, try Soul Folks. This link has info as well as other restaurant reviews;
There are some great museums downtown, and shows at the Ahmanson, or Mark Taper Forum.