Can you clarify that you're interested only in St. Paul, and not Minneapolis?
If that's the case, I'd second Brasa and Ngon, and Heartland for that matter (though Heartland is more of a fine dining option; the other two are casual and relatively cheap)/
Another of our favorite spots is The Strip Club, a meat and fish place in the style of an old speakeasy. Excellent local, grassfed beef and generally slightly innovative takes on classic cuisine.
You also might consider the Dari-ette, a classic Midwestern drive-in with typical drivein fare as well as Italian-American stuff and great soft serve.
And of course there's Mickey's Diner downtown, a classic divey dining car which has been open 24 hours a day, every day, since the 40s. Opinions are divided here at Chowhound on the food, though I consider it good diner food.
36 7th St W, Saint Paul, MN 55102
The Strip Club
378 Maria Ave, Saint Paul, MN 55106
\Welcome to St. Paul.
A few thoughts:
First, Heartland. Excellent food, all local and organic, in a restful setting. If you want something lighter, there are great choices in the adjacent wine bar. Definitely "local specialties." http://www.heartlandrestaurant.com/
Next, Brasa. Slow cooked rotisseried chicken, pork and beef. Interesting southern sides (collards, yams with andouille, cornbread). Menu emphasizes the local and organic. This is on Grand Avenue, a good place for a stroll after dinner. http://www.brasa.us/menus_eat.aspx
Izzy's for great ice cream cones. http://www.izzysicecream.com/
The Day by Day cafe for great breakfasts. http://www.daybyday.com/
128 cafe. A small cafe with great baby back ribs. A small menu that changes daily. http://128cafe.net/menu.htm
(Listmates -- I was having trouble adding the links for these via the site. Not sure if it's not working or just me.)
I forgot the obvious.
If you want to try local specialties . . . . you need to think about some good asian food.
This is home to many Vietnamese, Hmong and others.
Lots of small asian places on University Avenue.
One place we really like is Ngon Vietnamese Bistro. KInd of Vietnamese French. Everything very fresh. We like the pho there. Local special meats and a great local beer list. http://www.ngonbistro.com/
Also, there's the great barbeque place on weekends on the avenue. Help me out with that, listmates.
Ngon Vietnamese Bistro
799 University Ave W, Saint Paul, MN 55104