I'm not real familiar with the central and western part of Nebraska, though I have heard that Trattoria Flippin' Sweet in Kearney, NE was just named the best pizza in the state and one of the top in the midwest, by Midwest Living Magazine. So that might be worth a stop.
I'm more familiar with the greater Omaha area, having grown up there. My suggestions in that area would be:
(1) Eat steak. My favorite locally-owned steakhouse is Anthony's. My favorite chain is Flemings. Ask around and you will find strong opinions about steakhouses and I have found it generally tends to be loyalty to what you grew up with. Most of the locally-owned steakhouses will be similar in that they do steak excellently, have pretty similar sidedishes, offer ridiculous amounts of food, and are in dated buildings. The chains to be a bit classier and more modern in terms of decor and offer more updated sides. An authentic Omaha experience can be found at Anthony's. My second choice would probably be Piccolo Pete's.
(2) Go to the zoo. Even if you don't have kids with you, you won't be disappointed. This zoo is consistently ranked in the top zoos of the country year after year. They just finished a beautiful remodel of an already impressive aquarium....but there is more to the zoo than that. You can bring in a cooler with lunch if you don't want to pay high prices for concession stand food.
(3) While you're at the zoo, swing by Zesto's - just outside the zoo and treat yourself to the ice cream touted year after year on the College World Series.
(4) Speaking of, if you are into sports, and are coming through in June, check out the College World Series....it's a good time. I'm not even into baseball all that much but love checking out a few games each year because of the excitement and ambiance. Bring sunscreen & sunglasses.
(5) The Ashland, NE exit...roughly 426, I believe. There you will find probably the most modern/commercial of our state parks (Eugene T. Mahoney), The Strategic Air & Space Museum, The Lee G. Simmons Conservation Park and Wildlife Safari among other things. If you only have time for one thing at this exit, I'd go for the Air & Space Museum. I have been to plenty of aviation-oriented museums and I can honestly say it's impressive. While you're near Ashland, you might want to check out Parker's Smokehouse for BBQ...a lot of Omahans and Lincolnites will drive here for BBQ.
(6) The Old Market in Omaha unique dining and shopping options. My personal favorites down there are M's Pub for food and Ted & Wally's for homemade ice cream. The Old Market is not too far from other area attractions, like the Heartland of America Park and the Bob Kerrey Pedestrian Bridge.
(7) Dixie Quick's for breakfast in Council Bluffs, IA
You didn't say how far you were going into Iowa, or in what direction...but I'm familiar with a lot of Iowa, so if you post what part you are headed to, I might be able to give you some help there.
Another thing I just thought of in Omaha during the summer would be Rick's Boatyard. It's a restaurant/bar on the Missouri River just north of dowtown. The food is average, in my opinion but the drinks are great and so is the view. On weekends in the summer they sometimes have live entertainment.
Just an FYI - Rick's Boatyard closed at the end of the year. There are apparently negotiations with the city about reopening under new ownership, but there hasn't been any recent news.
Thanks for the heads-up, Suzi-Q...that's too bad they closed. Maybe something good can come out of it though.
Along I-80? We liked North Platte. The Golden Spike Tower gives you a bird's eye view of the Union Pacific Bailey Yard. The Buffalo Bill Ranch has his interesting barn and Victorian house. We clambered around on trains and looked at animals at Cody Park, missed the kiddie amusement park. There is also a 20th century veterans memorial just south of the interstate.
The Stuhr Museum of the Prairie Pioneer in Grand Island is good. We would have enjoyed it more during the summer season when the railroad town is open.
The Great Platte River Road Archway Monument in Kearney may be worth a stop. We didn't.
I agree with divinghkns' recommendation of the Strategic Air Museum and zoo.
Other opportunities abound if you are willing to get away from the interstate.
The Mormon Trail Museum in North Omaha is very nice and it is free. It is similar to the Oregon Trail Museum in Baker City, OR.
I'm from Kearney and I would second the suggestion on stopping to eat at Flippin' Sweet - it's AMAZING pizza. IF you are in Kearney on Friday or Saturday night or Sunday afternoon, check out what the World Theatre has playing (it's right across the street from Flippin' Sweet) - it's a beautifully renovated historical theatre that plays classic, indie, and family-friendly movies. Also, if you're in Kearney, you might as well check out the Archway exhibit.
I'd also suggest the Henry Doorly Zoo in Omaha - it's a wonderful zoo and you could spend a day there even if you don't have kids (actually... i prefer to go without kids =). My favorite part of Omaha is the Old Market - i love walking around and looking at all the different shops, plus there's tons of great food options from cheap to fine dining. We always stay at the Magnolia hotel (which is beautiful! and has extremely comfortable beds) so that we are in walking distance to the main Old Market area so we don't have to drive much.
Enjoy your trip... I'm glad you'll be seeing Nebraska as I think many people overlook it's charm ;)
I lived in rural Nebraska a few years back and (if you have the time and are in no rush) I'd get off the interstate, and drive the state highways (6,136,20 just to name a few) and pass thru some small farming towns, pleasant rolling scenery, stop for lunch at a local cafe and chat with the locals.
Also, if you can catch one of the county fairs most of the counties have during the summer, this will give you a glimpse of local life.
I've lived in many places around the USA, but I have to say that Nebraska and the folks from Nebraska will always have a special place in my heart. Kind, hard working folk with pleasant senses of humor. I'll always cherish the memories.