You could go up to Dundee and sit on the bench where Warren Buffet and Paul McCartney were hanging out the other day.
http://www.usatoday.com/story/news/nation-now/2014/07/14/warren-buffett-paul-mccartney-selfie/12636247/I have not seen the Old Market area mentioned - along Harney Street from 10th to about 14th on the South side of down town. For breakfast, I recommend Louie M's at 16th and Vinton a couple of miles south of downtown.
As with any city of its size, crime isn't limited to just the "bad" areas. Even in many of the areas mentioned, and particularly in the downtown areas, avoid walking around at night where there are not a lot of people around.
Couple other thoughts, if you are driving I-80, depending how fast you intend to get accross Nebraska and whether you have kids or not, there are a couple of things you can do along the way such as spend an hour or so walking through the Kearney Arch (pioneer museum that spans I-80 at Kearney - actually pretty well done but a little pricey at $10 or $15 I think), and having lunch or dinner at Ole's Big Game bar in Paxton (good enough food, but interesting atmosphere with trophies from a life time hunting around the world (unless you are of the anti-hunting crowd, then I dont' recommend it). If you have some time and want to avoid the monotony of driving I-80 along the Platte River Valley for hours, take Highway 2 from Grand Island west through the sandhills, but be prepared to see few cars - and watch your speed.