The problem with Nebraska is that farmers own all the land. Unless you know someone who is willing to let you ride on their land you're pretty much hosed for real dirt riding. Doubly so if your bike isn't a dual sport. Move to Arizona or California if you're serious. Colorado is an OK compromise but it's just that... Anyhow, It's a three or four hour drive in any direction for any sort of public riding. The Headworks OHV park and River Valley OHV (South O Bridge) are both closed indefinitely. There are more than a few private tracks and trails but riding without trailering in the immediate area without networking is down to Exploration or Hooliganism. I suggest the former as it the adult option involving more respect and responsibility with less d-bag-ness.
Some Useful Links:
There's also Timber Ridge but it's a community/family membership situation so play nice and bring your checkbook:
Networking on community venues and dealerships/shops will open many options too.
Dillon Brothers, Edwards, Star City, Etc
Dillon Brothers, Edwards, Star City, Etc
In closing: I have several dirt/dual-sport bikes and I almost always end up loading a couple (plated or otherwise) in my truck and driving for hours if I want to spend a full day riding dirt.
[Missouri, Kansas, Iowa, Colorado, et al.. Literally every state bordering us has great options. The only respectable OHV Parks in NE are a minimum 3.5 hours out and at least one is nearly all sand trails so choose carefully.]
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Abbott is in Lincoln, so about 1 hr. away.
There's one in Council Bluffs but it's closed due to the March flood.
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https://www.riderplanet-usa.com/ is the best source for OHV trails.
The one up in Mapleton, IA looks pretty cool, but I think the membership price was quite steep.