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.
Have you looked over the chaser guides on here? I have had good luck using them. As the others have said you gotta visit the Negro League museum. When I was there they had a jazz museum there too. I enjoyed both!
Go Downtown to the National World War I Museum and Memorial across from the old Union Station. Both worth several hours. Only go to Arthur Bryant's if it is an off-day and never on the weekend. Lines can get discouragingly long. Great food though.
We sat by the fountains and loved it. Definitely go to the Negro League Baseball museum. My boys also enjoyed the College Basketball Experience!!