The other posters pretty much hit the nail on the head as far as tires go. I personally like Bontrager Jones AC's, but everyone has their own preferenecs. The key is to run a tire that is made for dry weather conditions. We don't really have many roots or rocks, so traction is really not a big deal out here.
Also because of our soil type, wet-riding conditions majorly suck and I avoid them altogether, so you probably won't need anything that is designed as a mud-shedder. Great if you live in Pack NW or New England, but not very practical here.
We live in the land of hard-packed soil, so no need to put a tire on your bike that is designed to handle roots, rocks and mud.
Places to ride in Eastern NE? Well, the best rides have already been mentioned: Platte and Lewis and Clark. But Indian Caves, Ponca, and Stone State Park are also really nice. Ponca and Stone are near Sioux City and you can make a day and ride both. They are both really good rides, and you can camp there.
Then you have the great Omaha "neighborhood" rides: Manawa, Tranquility, Swanson.
Lincoln folks have Wilderness, Branched Oak, and Pawnee as their neighborhood rides, as you probably know.