For hot dogs:
Hot Doug's - They flip the bird at the Foie Gras ban and make kangaroo dogs with basil remoulade. Fridays and Saturdays only, you can get their famous duck fat fries, which are worth the trip alone.
Weiner Cirlce - Not only do you get the sassiest service in town but you get the best traditional Chicago style dog. Get a char dog with everything and fries. Sit at one of the picnic tables out front. A true Chicago experience.
For deep dish pizza - Seriously, deep dish pizza is disgusting. It will make you feel gross for days. Not only that, but all the Chicago deep dish pizza restaurants are always packed with tourists and have lines hours long. For really really good pizza, go to Damenzo's on Taylor Street. You won't be disappointed. If you really have your heart set on deep dish, I've heard that there is a pizza place in Morton Grove called Burt's that is less "deep dish-y" than your standard Gino's East style, and comes highly recommended. Otherwise, put on some elastic waist pants, grab some Tums and head to Gino's East.
Don't forget that Chicago has a huge Mexican population and it can be some of the most delicious AND cheap food in the city. I really like Tecalitan (mostly for the salsa) but La Pasadita is cheap. For a really good Mexican meal, go to Fonda Del Mar. It's a more modern, fresh take on Mexican cuisine with the best Pork with Achiote Mole I've ever had.
One more suggestion: there are a ton of amazing restaurants that offer prix fixe meals on Mondays and Tuesdays for cheap cheap cheap. Check Metromix.com for them. I know Le Bouchon (traditional French) is a great date spot.
Have a great time!
I've lived here for 11 years now, and I still don't get the deep dish thing. It's mushy, not very tasty, and just too much of everything. There are many threads on here about best local cheap grub. This week's Chicago Reader has a whole section on great sausages of Chicago, from hot links to German style to Thai renditions (www.chicagoreader.com).
A couple of places that you should try in the under $10 range are Cafe Iberico (tapas, great sangria...two can eat and drink til you burst for about $40 or even less), Spoon Thai (with its well-known "secret" menu), Tank (Vietnamese). Also, if you go to Chinatown, you can get incredible dim sum for about $20 for two. Search the board for dim sum; there are several that are recommended.