The big claims to fame are horse racing and college basketball. A UK basketball game is the thing to do if the timing is right. You are too early to see horse racing at Keeneland, but you could visit the KY horse park.The area around Lexington offers lots of hiking (Raven Run, Berea Forest, Natural Bridge) and canoeing (Elkhorn Creek and Red River)--but weather is definitely a factor for January.http://www.kyhorsepark.com/http://www.ukathletics.com/sports/m-bask��
The university campus near the center of town is quite interesting. The library is an architectural extreme that has to be seen from the inside (top floor). Also, there are some reasonable eateries��no lard nor bacon��across the street from the university.
You can go to the Kentucky Derby. But,,,since that happens only once a year, you might have a bit of a wait.This answer gives you NO more indormation than your question.