A few places that Ive been: Z-Harvest, just off of University is one of the only "non-townie" places that i know about. I've eaten there a few times, and its quite decent. No fried cheese curds on the menu! A sophisticated menu that varies from nice sanwhiches to main entrees, and includes veg options. The people who own it have been running the best restaruants in the area for 2 decades. Its a very small place, so call if you've got a big group.
There's also a place (that my mom says moved recently) It was on the east side, on the corner of Main and Libal, just beyond East High School. Very nice place for soup, sandwiches.
If you're near Lambeau, try Sammys. Great Pizza. In West DePere, just across from St. Norberts, there's also a really good pizza place, with an Irish name... (i forgot the name)
East DePere has really changed a lot (Lived there for 17 years!) Theres a wonderful little international style bakery that opened where "The White Store" used to be. I was there last year, and it was impressive, especailly for someone who avoided the Swan Club like the plaque. Rumor has it, Swan Club has re-invented itself....somewhere in DePere. Seroogy's chocolate is amazing too. I make a pilgrimage to DePere, just for Seroogys.
In general Green Bay has a lot of good mexican. Los Banditos is an old standby, but because theres so many Mexicans in the area, there are lots of small places that i hear are good.
Storheims has the best custard on THE PLANET. Its amazing. All tourists should try Storheims. Ice Cream will never be the same. Their buttered hamburgers are also an acquired taste, that I avoid.
Otherwise, go up north about 45 minutes, and check out Sturgeon Bay. Great Eats in Door County!
As you may sense from from my screen name, I grew up Green Bay (and I still haven't recovered).
It's been a couple years since I've been up there, but the Union Hotel in De Pere (just south of Green Bay on the East Side of the Fox River) is an landmark with fine steaks.
Here's a little info:
If you want to go cheesier, which is probably what you'd want to do, there's Kroll's West for butterburgers in the shadow of Lambeau Field.
And there's always Brett Favre's steakhouse.