I have traveled along your route a lot and only live 2 1/2 hours south of you, so you are going to want to print this for recommendations. I journal our trips and am looking at that. These estimations do not include stopping to sit down and eat, but do include quick bathroom stops and fast food. It will take you approximately 3 1/2 hours to reach Salina, KS, 5 hours to Wichita, 6 1/2 hours to Oklahoma City, and 8 1/2 hours to DFW. After that, we did a lot of poking around so I really couldn't say how long it will take to get further south! Of course all this depends on the time you have and your interests, but here are some recommendations to see to or fro.
Salina, KS - Rolling Hills Zoo located NW of Salina on I-70. The cleanest, most beautiful and spacious zoo I have ever seen.
Near Lindsborg, KS - Coronodo Heights is an amazing lookout hill where Coronodo visited to explore
Oklahoma City - The federal building bombing site is moving.
Pauls Valley area, OK - I think this might be a good stopping for you after about 8 hours of driving. Next day, go to Turner Falls State Park where the kids can swim/play in a waterfall. I do remember the entrance fee is a little pricey. Also, in this area are small amusements like bumper boats, a wildlife safari, and a lake. Also, I can't remember where it is between Oklahoma City and the TX border, but there is a small wildlife rescue type zoo that was neat. You will see signs for it. The owner even came and gave us a personal tour and food to feed the animals and allowed the kids to pet some.
OK-TX border - Lake Texoma - HUGE lake and resorts
Fredricksburg, TX area - NOT TO BE MISSED IN APRIL! Between Fredricksburg and New Braunfuls is gorgeous hill country which will be in bloom with bluebells and indian paint brush while you are there! Gorgeous country drives, Lyndon B. Johnson State Park, wineries, great German food, and wonderful quaint shops at Fredricksburg.
San Antonio- I agree with the other posters. Besides the Alamo, there is the River Walk, Sea World, Six Flags, and caverns to tour.
Port Aransas, Rock Port and Mustang Island - I would highly suggest staying at Mustang Towers on Mustang Island. Find information on line and reserve in advance! These are owned time-share condos to rent (no tour required) right on a long wonderful sand beach that the kids would love! Bring or buy sand toys! There is a swim pool at the condos. You can even drive on the beach. If you are camping, stay at Goose Island State Park or you can camp free to cheap right on the beach at Mustang Island. Plan to visit Goose Island State Park. Unless the signs warn of red tide, you can pick oysters up right along shore and take them back to the condo or camper and grill them!
Things to do, see, and eat in this area:
Corpus Christi has a tour of the USS Lexington ship. There is an aquarium but I have not visited it.
Rockport, Port Aransas, and Goose Island: Visit the 1,000 year old oak tree. If you like fishing, take your poles and tackle! Permits are cheap and you can fish just about anywhere in this area. There are wonderful piers to fish from as well. Also, go deep sea fishing with a company called Dolphin Docks. I have traveled all over the caribbean and Florida and the best deep sea fishing ever was with Dolphin Docks! It was a hoot! They will clean your fish and you can pack it home in dry ice. Here's what I recommend if you do that: Wear old clothes as you will be a stinky mess after the day. Plan to bag them up tight in a garbage bag or wash them right away! Eat a good breakfast so you don't get sea sick. Have dramamine for the adults and liquid nausea medicine (very cheap over the counter) for the kids. If the little ones start looking sleepy or get sick, give them a swallow of the liquid nausea and it will bring them around. I can't recommend taking the 1 year old on an all day trip, maybe 1/2 day trip. There is a short ferry ride between Port Aransas and Rockport which is a nice experience for the kids. You can ferry your vehicle. You can get around this entire area fairly quickly, but remember Texas is a huge state and everything else throughout Texas is further than it looks on the map!
Eats: I journaled that the "Seafood or Shrimp Shack" was good fast food, at Mustang Island I think. The Quarter Deck was not a very good value. Charlotte's at Fulton was awesome food. Eat on the deck!
Weather: You're from NE, so am I originally. You won't notice the humidity! South of Hill Country you will do fine with summer shorts and capris and maybe a jacket at night. North of Hill Country, I would have some long pants and layers. It can be windy on Mustang Island, everyone might need sunglasses to protect eyes from blowing sand. Have a great time!