For a simple lunch, I have always loved the Classic Cup. Not too terribly pricey either.
For a good steak that is reasonable, I found the Plaza III steakhouse to be very good. Starker's Reserve also prepared a very good steak that was reasonably priced. Edit - Starkers Reserve has closed. Apparently the Head Chef passed away.
As was mentioned, the Capital Grill is good, but is a little steep, and I believe is a national chain also. The steak was great, but my favorite at the time was their jalapeno cole slaw. Bonefish is also a chain, which you can have in Omaha, so I would stick to the locally-owned places. STAY AWAY from the Hereford House, unless you want the most bland steak you've ever had.
There was a place called Boozefish on West Broadway that I loved. I believe it's still there. My all-time favorite was Lidia's, close to Union Station. It didn't break the bank and the owner/chef is/was well renowned. Pierpont's in Union Station was also very good seafood. Piropos, which is north of the river, I thought was quite good also.
Speaking of things to do, Union Station usually has an exhibit of some sort taking place. Science City is great for kids.
http://www.unionstation.org/
Succotash for breakfast was a very cool experience. Blue Bird Bistro is good as well. Both are just a really fun experience. Succotash is sort of South of downtown and Blue Bird is West of downtown.
There is a place in Westport where you can get a very cheap hour-long clothes-on full body massage for about $40 apiece. It was a lot of fun, because it isn't a private massage, and probably 15 people are getting a massage at the same time, so it's sort of a massage collective. But it is very relaxing after dinner, and they will put you side by side with your spouse or significant other. It's pretty cool and fun actually.
The Kemper and Nelson Atkins Museums are pretty cool. Have fun..