Well, I live in Lexington, so here is what you need to know...
Stay in the Nicholasville Road area if you want to be around the mall and close to the stadium.
Downtown has some hotels, but not a ton and the prices will likely be high. A lot of really good places to eat, drink, etc are downtown, however, and is my favorite place to hang out for a fun evening. All of the stuff downtown is basically locally-owned, and the restaurants are fantastic for the price. A lot of local breweries here as well, with fantastic beers.
Hamburg area has a lot of shopping and dining, but you will be a little away from other parts of the city.
We just got UBER taxi service, along with other regular taxis, so getting around isn't hard. The bus service here is good, with a lot of stops and they are pretty much on time (every 15-20 mins per stop). But if you give me a heads up, I can help you out a little bit with getting around and maybe grab a drink or three. My family is also coming up, so a few State fans will be around for some fun. If you stay downtown, the city bus service has a shuttle that takes you right to the stadium from Rupp Arena.
Keeneland is fantastic, bring a lightweight suit if you really want to experience it. The ladies really doll up and offer some great scenery between the ponies.