If you go to a Durham Bull’s game, go in a Saturn. They will usher you into your own private and free parking. No problemo. The Bodies Exhibition was great, but I don’t know if kids would enjoy it that much and it’s $24 to get in. If you are into racing, the Richard Petty Museum is in Randleman (on the Petty’s property). It’s a bit of a drive from Raleigh, but worth the effort if you are into that sort of thing. There are plenty of sports bars in the area that will be showing the race. Damon’s is where I would go. If you want to beat the heat and go ice skating, there are several rinks that have public skate schedules. Cary and Garner Ice House, The Rec Zone in Raleigh (where the Carolina Hurricanes practice), and The Factory in Wake Forest. If you google the sites, you can find their schedules. Danny’s Barbeque is awesome. The 25th there will be a Bud Light Downtown Live - free live music with the headliner band that week being Warrant (they will go on around 8 pm). You can find out more about that by googling “96 Rock Downtown Live”. The flea market at the Fairgrounds is great, but go early. I’ll be hotter than Satan’s anus that week. If I can think of anything else, I’ll post it.
For the kiddie, try Playspace museum
http://www.playspacemuseum.com/ although it will only be in its current location a few more weeks, no doubt the new one will be even better
and right nearby is Kazoom Puppet Theatre, very appropriate for 2 yr. olds, on Wednesdays and Thursdays
http://www.carolinapuppets.com/kazoom/index.php
If you do go to the NC Museum of Art and it is bearable outside don’t miss the “Cloud Chamber” http://ncartmuseum.org/collections/highlights/20thcentury/newcommissions.shtml
If you are in the shade, the heat is not so bad, there are many, many hiking trails everywhere
http://ncnatural.com/TriangleRec/HikingTri.html
I just checked the cast list. Lots of good people in there. It should be a good show, if you’re interested.
To be honest, the name scares me. 
Great suggestions all! I’m printing things as we speak to go into the trip folder.
The review says, “A very good show with a very bad name.”