I now live in NC but started sharing the Avetts' music with friends back in NE several years ago. A few favorites from my hometown:

Best cool kids' pizza & beer: Yia Yia's (http://yiayiaspizza.com)

Best dive: Original place burned down, but I understand Norm's still serving a mean Italian Beef at M&N Sandwich (http://www.facebook.com/MandNSandwichShop).

Best Indian: Haymarket location of The Oven (http://www.theoven-lincoln.com). Have the mulligatawny and look for the picture of Bono. Excellent overall experience.

Finally, nothing says "I've been to Nebraska" like an original Runza Sandwich (http://www.runza.com/menu). Runza is a regional fast food chain that's a bit bland overall, but is unique for their namesake stuffed sandwiches of the bierock/pirogi tradition.