To me, downtown LA (the main downtown section) is a place to work and if you have tickets for a professional game to go to The Staples Center. There is a little more to offer, but not much. There are some interesting places to eat, like The Pantry, but there isn't much else here expect a few tall buildings and a bunch of bums (you could have some really interesting conversations here). There is a walking tour of some of the famous and historic buildings, too, which could be fun...Bunker Hill, The Fine Arts Building, etc.
Maybe you would want to stop by the Jewelry District if you want to buy jewelry, but downtown isn't really the place to be for tons of action and fun.
Near downtown is China Town and Olvera Street (see must see section), which are cool.
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Alternative: Go to the many other options around LA...Hollywood, Beverly Hills, the mountains, the ocean, etc.