This is by no means. AT ALL. A complete guide to Chicago. It is a guide to my small red-line slice of it, my stompin' grounds.
Know where you're going to eat, if you plan on eating out. Chicago's gold coast/loop area is 99% either fast food or super-expensive gourmet restaurants. You will not find a lot of inbetween.
Bring a compass. Except for a few crazy diagonals (Clark, Lincoln) all streets run north-south or east-west. The lake is east.
Okay. Starting from the north...
Red Line: Sheridan stop. The northern boundary of Wrigleyville, the corner of Sheridan and Irving Park. Head north to Irving Park (that's a left as you leave the station) and then turn left again. Byron's Hamburger's is half a block west. Delicious burgers and hot dogs, cheap, and great fries. Try the onion straws, they're amazing.
Keep going west on Irving Park and you will hit two cemeteries at the intersection of Clark and Irving Park. On the north side is Graceland, which has many historic and beautiful tombs and headstones. On the south side is Wunders, less well known but no less beautiful. (Christopher Manuel is buried in Graceland.)
Red Line: Addison stop. One stop south from Sheridan, the official El stop of the Chicago Cubs. You can see Wrigley field from the train. Which is fun!
Gold Coast is the area south of the North Side but directly north of the downtown "loop". It is predominantly a shopping and tourist district, intermingled with a business district. There's lots of fun stuff to see here.
At the Chicago and State stop, you can either walk east about four blocks (easy directioning: McDonalds is at the northeast corner of the intersection) or catch the eastbound 66 bus. Either will bring you to the Water Tower, one of the only surviving buildings from the Great Fire. Oscar Wilde called it the ugliest building in North America, but I rather fancy it. It has a tiny museum inside it.
Northwest of the Water Tower is the Hershey Store, which is fun. Northeast is Water Tower Place, with lots of shopping and a Wow Bao stand in the lower level. Another block north of Water Tower Place is the John Hancock Centre, which is fun if you like being up high in tall buildings. Apparently it's more fun than the Sears Tower though it doesn't have new glass-floored observation porches like Sears Tower does.
If you walk south down Michigan Avenue from the Water Tower, you will pass a ton of interesting shops. After two or three blocks you'll hit the cross-street Ontario, with Grand Lux on the corner. Grand Lux has really great food and is a fun place to eat, but if you're hungry and the wait is long try Lawry's across the street, especially their lunch sandwich bar. Heading further west down the street you'll hit Trader Joe's, which is an awesome gourmet/organic grocery store. Across Ontario from Trader Joe's is Pizzeria Due, which supposedly has good Chicago-style pizza. As I loathe Chicago-style, I can't vouch for this :D
Two blocks south of Ontario is Grand Avenue. You can walk down some steps from Michigan to get to it, or take the red line to Grand & State.
NOTE: One of the best cost/comfort balances in a downtown Chicago hotel is the Hampton Inn, a block south of the Grand&State stop at Illinois and State (just off State, actually, to the west). Cheap quality rooms, free breakfast, very close to everything.
At the Grand & State stop, on the northeast corner, is the Weber Kettle Grill, which at first looks a little expensive but gives you a LOT of food for your dollar. Sit at the Chef's Table and watch the grills go! Excellent grillfood and one of the few places in Chicago that I trust to do me a medium-rare burger.
Two blocks east of Weber Kettle Grill, on Grand, is Oysy Sushi, delicious and reasonably priced. Keep going east another two blocks, turn left and go up the steps, and BAM. You're back on Michigan Avenue, right outside Northbridge mall (Lego store on the 3rd floor, otherwise unremarkable). If you were to keep going on Grand, you would hit Navy Pier -- more on Navy Pier in a minute.
Travel south on Michigan from Grand and you will see the gothic Tribune Building on the left, past which is Pioneer Plaza. The Trib Building has a ton of neat shit embedded in its exterior, including elements of historic temples, Roman baths, the Sydney Opera House, and a piece of twisted steel from the wreckage of the World Trade Centre (south wall, about halfway down, just past the entrance to the radio station).
Pioneer Plaza is the location of Jean-Baptiste Pont du Sable's original dwelling; he's considered the founder of Chicago. Other than a plaque proclaiming this it's unremarkable except for an adorable and anatomically correct moose statue (permanent) and a hideous sculputural representation of American Gothic (hopefully temporary). Pioneer Plaza faces the Wrigley Building, also quite nice. (If you're desperate for food and go through the doorway under the sign reading WRIGLEY BUILDING, and pass through the courtyard there, you will find the Super Secret McDonalds.) Just past the Wrigley Building to the west is Trump Tower, which is...big and shiny, I guess.
Any further south on Michigan would take us across the river, so put that on pause for a minute.
If you come out of the Grand & State red line stop, go one block south to Illinois. There's a bus stop there where you can catch the 29 or 36 bus, or the blue free-shuttle, to Navy Pier. You can walk, of course, but it's quite far. It's about a $5 cab ride if you can flag one down. Just tell them Navy Pier, they know where it is.
Note: The buses will all take you past Fox&Obel, a really really good gourmet food shop. The restaurant at Fox & Obel is shit, however, don't eat there.
If you are going to Navy Pier, you will find that 99% of it is tourist traps. They do have an awesome ferris wheel and a gorgeous stained-glass museum; pick up a map at the entryway infobooth. Chicago Shakespeare also makes its home there, but it's super-overpriced for the quality of theatre you generally get. There's also an IMAX right near the entrance to the pier, if you want to see a show (I prefer the AMC River East, on Chicago just past Michigan).
The best part of Navy Pier and the only reason I ever go there is the Shoreline Sightseeing Tours "River Architectural Tour". One hour on a boat, going up and down the Chicago river, with a guided tour of the riverfront architecture. It's the best tour in the city and one of the most pleasant hours you can spend here. It's about $25 and worth every penny. It boards just southwest of Navy Pier, but you buy your tickets at a stand on the south side of the pier.
Okay, that's Gold Coast.
THE LOOP. OH THE LOOP. So much to see, so much not to see.
If you're heading south on Michigan, you'll cross the river. Eventually after about two blocks you'll hit Millennium Park (where the Bean is), then the Chicago Cultural Centre (across the street -- go inside and poke around, it's pretty) and then the Art Institute, which is glorious. Keep going and you'll end up at Grant Park. Incidentally, Buckingham Fountain in Grant Park is a pretty place to hang out and the location of the penultimate scene in Nameless.
If you'd like to see more loopishness for your loop, take the Red Line, which will let you out at several points on State Street. To see Daley Plaza (location of the Picasso Baboon and also of Torchwood Chicago) take the Red Line to Lake, go one block south on State Street to Washington, and then go two blocks west, it'll be on the right. Across from it is the First UMC of Chicago, which has some pretty sculptures. Backtrack to State Street and look north to find Marshall Field's, now Macy's, which is a cool place to shop.
If you take the Red Line to Jackson, walk a block or two south (depending on where you leave the station) and the Harold Washington Library will be on the right. It's fun, but disorganised. Walk east from there down Congress and you'll come to a Panera restaurant, across from which is a really really good hostel. It's expensive for a hostel ($35/night) but very safe and really nicely appointed. The last time I stayed there they didn't have WiFi, but the Panera does.
If you take either the Red Line to Jackson or the Brown Line to Quincy and Wells, head west to reach the Sears Tower just before you'd cross the river. You really fucking can't miss it, promise. :D I've never done the Sears Tower but I feel like I should, soon.
There is a lot more to do in the Loop than I'm mentioning, but in the past decade or two focus has slowly shifted away from State Street in the Loop, and more towards Michigan Avenue in Gold Coast.
I don't have much experience with the South Side. You can take the red line to Cermak-Chinatown and bum around Chinatown; take it to Roosevelt and walk east to get to the Field Museum, the Shedd Aquarium, and the Planetarium. You can also catch the Water Taxi to the Museum Campus from either Navy Pier or the lower deck of Pioneer Plaza. The Water Taxi is a hoot and fairly cheap, I think about $5. (It's $3 from Pioneer Plaza to Union Station, where the Amtrak lives.)
You can also take the red or green lines to the Garfield stop and catch the eastbound 55 bus, which will take you past University of Chicago and to the Museum of Science and Industry. The MSI has a Harry Potter exhibit running right now; I've been and it's excellent. On the University of Chicago campus, check out the Du Sable Museum of African American History, the Smart Museum of Art, the Oriental Institute Museum, the Seminary Co-op (aka 57th Street Books), and just past the Seminary Co-op "The Med" -- Medici restaurant and bakery. They do awesome hamburgers.
OW MY BRAIN IS TIRED YOU GUYS. So I'll stop there. Please feel free to add information -- I obviously stick to one area of town and have very little to offer in the way of 1. things you can't get to via public transit or 2. the south and west sides of Chicago.