Being a small stadium I found that they might have the best cheap seats around, if that makes sense. Even the upper deck feels close. Get there early, the tail-gating scene is arguably the best in baseball. Also take a tour of their museum.
In town, you MUST go to the Negro League Baseball Museum. Can't miss. Down the road is a small monument for where the Monarchs used to play.
Barbecue. Make time for Joe's (the Z-Man sandwich and burnt ends), Arthur Bryant's (ribs and burnt ends, one of the first barbecue places in the country), and Gates.
Take a drive by the downtown Library if you can as well.
I can second almost everything Paul said.
Definitely plan some time for the Negro League Museum! It is a pretty dense, information intensive exhibit, so plan to take it slowly.
Another good BBQ option is Fiorella's Jack Stack. If the weather is nice, I recommend the patio.
I'm not always a bleacher kind of guy, but loved the seats out by the fountains, especially since it was a pretty hot day when we went. The lower level infield seating area feels amazingly close to the field, too. This is a gem of a ballpark, and in my opinion pretty underrated.
The tailgating was okay. . . but I'm from Milwaukee, where tailgating is a whole different level event.
Milwaukee is it's own level of tailgating. You know it's good when there's an on-site butcher shop.
They don't really tailgate here in Texas any more. I was impressed with how they organize the parking (here you can just drive up and down aisles). Once an aisle is full, they just take it over. For baseball I thought it was great. Obviously Chiefs games next door and college games are another level.