My personal opinion is to stay in Lincoln, if financially possible. If the goal is to get the full experience, then you're not going to get it an hour away in Omaha.
Friday night, go downtown, walk around campus, soak in the atmosphere, go to downtown or Haymarket bars/restaurants. Go to the stadium, see the statues and trophies without the Saturday crowds, etc. Maybe even find a group of fans from this site or others that are meeting up or tailgating.
Saturday, get downtown as early as possible. Again, tailgate, visit bars, walk campus, etc. to prime for the game. Depending on your seats, get there an hour early. Watch the warmups, band, etc. After the game, hang out in the stadium for a bit to soak in more atmosphere. Then, depending on the time of day, walk around campus, downtown, Haymarket, eat, drink socialize, etc. The popular places have undoubtedly changed since I was a Lincolnite, but I had season tickets for 15 years and the pre/post-game experience was half the fun.
No offense to the Omaha folks, but if someone asks me what to do in Las Vegas, I don't suggest they stay in St. George, Utah because it's cheaper.