As to museums, I have several recommendations for you! Here in the D/FW area, my must-see museums are the Cavanaugh Flight Museum in Addison and the Frontiers of Flight Museum at Dallas Love Field. Cavanaugh hosts a great collection of warbirds. Frontiers of Flight covers a broader spectrum, including some amateur-built, and does a great job of showcasing the rich aerospace heritage of the D/FW area. Both are outstanding.
For a more out-of-the-way adventure, go to the south end of the D/FW area to Lancaster. The local wing of the CAF has a hangar there with several nice planes. The atmosphere there is very relaxed, and the CAF volunteers are very friendly. My son got to sit in their L-5 and make airplane noises and went home with a huge grin on his face. Just down the street is Air Salvage of Dallas. Their building and property is filled with all kinds of interesting relics. Warning: you may walk out with parts! The proprietor has been a big supporter of the EAA chapters here.
I haven't had a chance to see it yet, but I hear nice things about the Vintage Flying Museum at Meacham in Fort Worth.
For hobby shops in the D/FW area, I would drop into Mike's Hobby Shop in Carrollton. It is HUGE, has tracks for R/C cars, and all manner of R/C airplanes hanging about.
I don't have any aviation-related burger joints in mind, but I can set you up with all kinds of other good food in D/FW if you're interested. I like to take my kids out for picnics where we can watch arrivals at Addison, Love Field, and DFW. If that is your cup of tea, I can point you to some good planespotting locations.
On your way through Wichita, I highly recommend stopping at the Kansas Aviation Museum. It is smaller than the others, but very worthwhile. My wife indulged my request to stop there last summer on our way home from AirVenture, and even she was glad we did.
Enjoy the trip!