I will gladly offer you my high fives for free, assuming you're not coming before Sunday when I move back. That's good deal. I usually charge pretty heavily for a solid high five.
The only absolutely essential thing you have to do is get real Chicago deep dish pizza. I haven't been to the place Haschel likes, but if it's authentic it's legit. Other recommendations, in decreasing order of personal preference are Lou Malnati's, Pizzeria Uno, Giordanos, and Edwardo's.
Other restaurants with the Graebeard stamp of approval are
Urban Belly which amazing Korean food that's affordable-for-Chicago.
Macku which is the best sushi I've had anywhere, including New York, San Francisco, Portland, Seattle, and Vancouver.
Cumin, which is Indian/Nepalese. Not cheap but cheaper than others and a great value. If Indian is your thing you can head up to Devon street and walk around where vendors are selling mangos on the street and none of the signs are in English.
Kuma's makes a burger so good the only thing left of you after you spontaneously combust will be a glob of green slime. Bring your earplugs if you don't like metal. Seriously. Also, no minors after 10:00, which means you should get there before 6:30 since 3 hour waits are not unheard of. Don't worry, it's worth it. Best time to go is right when they open for lunch. I also have to recommend
Hot Doug's, even though it also always has a stupid long wait. Last time I went for lunch it was 1.5 hours waiting in a line around the block. But hey, the fois gras dog and duck fat fries were freaking delicious and there are no two finer words in the English language than "encased meats."
I also recommend ditching the car and taking the L around the city if you're staying within walking distance of a stop. It's a great way to see the city and will cost a fraction of the cost of parking. You can hop from neighborhood to neighborhood on the train pretty easily.
On that note, spend some time getting to know a neighborhood. I recommend Wicker Park. It's right off the blue line, has lots of food and < 21 entertainment options. Check out
Wormhole Coffee. It's a couple blocks from the L, has a full size Delorian, a usable NES with lots of games, and some of the best nostalgic posters and decorations I've seen in one place. Oh, and they make a hell of a coffee.
Museam-wise I will second the Shedd and the Museum of Science and Industry. The Shedd isn't as good as many west-coast aquariums, but it is a blast. The Museum of Science and Industry takes it's name pretty seriously. More actual science about industry than any other museum I've seen, if that floats your boat. The Chicago Art Institute is world class. If you're into art this is a must-see. Tickets are pretty cheap if I remember correctly, and it's right downtown.
You'll be here in the summer, so you have to check out the lake. Grant Park (and Millennium Park, the one with the Bean) is right by the lake and is great fun. You can also go north a bit to North Avenue Beach for actual sand and probably some volleyball.
There's lots of classically "touristy" things to do. I'd avoid Navy Pier. It's just an overblown carnival, and not even that great a carnival. Walking along the "magic mile" on Michigan Avenue can be fun. It's most fun when you've got lots of cash to blow on consumer goods, but it's also useful for getting that wonderful big city vibe of being shoulder to shoulder with the mostly washed masses while you look up at the great architecture. Speaking of architecture, there are several architecture tours. I can't recommend these highly enough, particularly if you're anything of an architecture geek. We don't have the greatest postmodern stuff, but if you're looking for historically influential architecture Chicago is your place.
There are some pretty solid venues if you're looking for nighttime activities. The Music Box Theater is lots of fun. I think they still do the occasional midnight showing of the Rocky Horror Picture Show. The Vic has some good bands. Blue Man Group is a staple and worth a viewing.
Too Much Light Makes the Baby go Blind is freaking awesome, and admission might be pretty cheap if you're lucky. Lots of comedy clubs around North and Wells, including Second City (lots of SNL alums) and Zanies (may be 21+, can't remember).
So, that's probably too much info. I could keep writing about Chicago ad nauseum. Let me know if you want more stuff, or send me a PM and I can give you my contact info.