I'm finishing up my undergrad here in Lexington, and there isn't a better time to be coming to UK, so congrats on UK Law!
As far as apartments, it all depends on what you want. I wouldn't recommend living too far off campus because, as someone stated before, traffic is terrible around town. Richmond Rd and Harddosburg Rd aren't as bad, but stay away from Nicholasville Rd. I'd recommend living around campus or downtown.
If you are looking into campus area apartments, there are a lot of new complexes opening up. Royal Lex and Newtown are close to campus (definite walking distance), but Royal Lex has been known to have break-ins and is more of a 'party' or undergrad atmosphere. Newtown is just down the block and it's very nice (only 4 years old), but moderately expensive ($600+/mo) and parking is a pain in the ass there (forewarning, parking is a pain in the ass everywhere in Lex/at UK). Other good places around campus: Red Mile (20 min. walk, well-priced), Campus Court (next to Red Mile, new), and Centre Court (expensive but great location, brand new). Also, the Lex is a new facility and more are being built every year around campus.
The Village is nice but it's far away, you'll need to buy a parking permit and those can be hard to come by.
If you're thinking downtown, check out Main and Rose (expensive but brand new) and Park Plaza (really nice, relatively inexpensive).
Good luck finding a place! You're going to love Lexington!