There are several good restaurants downtown, and a few that I have been to and can recommend. I like Saul Good and Village Idiot (both sort of pub food), Table Three Ten (small plates, charcuterie, seasonal menu), and Alfalfa (uses local products, good for vegetarians). A couple others that are close to downtown are Stella's Kentucky Deli (uses a lot of local products) and Nick Ryan's (steakhouse).
While we're on the subject of restaurants, I'll mention a few of my other favorites in town. Lexington has a fair number of good local places. For breakfast, I like Doodle's (standard breakfast fare, outdoor seating when the weather is nice), Great Bagel (bagels and bagel sandwiches), and Wild Eggs (if you are from Louisville you may recognize this one). There are also multiple places with good donuts, my favorite is probably Spalding Bakery.
There are a lot of restaurants near Good Samaritan Hospital, which you do some 3rd/4th year rotations at. I like Sav's Grill (West African, owner is super nice) and Han Woo Ri (Korean). Ellos is a good hole in the wall Mexican place. Dad's Favorites is a local deli; two locations (one of which is downtown) and only open for lunch M-F so a little hard to get to, but really good. Blue Door Smokehouse has excellent barbecue, highly recommended. Masala is my favorite Indian restaurant. Gumbo Ya Ya and Bourbon n'Toulouse both serve good Creole/Cajun food. La Petite Creperie is a casual French restaurant.