First off, this city is great for cheap things to do.
Cubs tickets: We always get them about an hour before the game. You can usually talk them down and if you aren't concerned about where you'll be sitting you can generally get a decent deal. There are various vendors that sell the tickets either in buildings or have a set-up on the sidewalk around Wrigley Field. In the worst case-scenario, you'll never pay face-value or watch the game from a bar across the street which will give you an experience. There are also walk-ups along that area that let you watch the game from the top of them. I've never checked that out, but they have food and drinks included in the cost of sitting on their roof-top bleachers.
Lincoln Park is lovely. There is a free zoo, conservatory, and lily pond. The nature museum is also in the area and I think it's $7. The butterfly room is worth the cost, the rest of it is sort of sub-par.
Millenium Park is fun. The shiny bean, the huge faces that spit water, flower gardens, a place to sit and relax while dipping your feet in cool running water, and if there is a festival it is generally there or in the adjoining Grant Park which has Buckingham Fountain. I love going there and taking work breaks since I'm right across the street from it. I feel lucky.
Navy Pier is fun. I skip the ferris wheel and ride the swings. They also set off fireworks on Saturday and I believe Wednesday nights between Memorial Day and Labor Day.
The lake shore walkway is very pretty especially when all the sailboats are out. It is a nice way to walk from park to park.
Food:
Giordanos has good pizza which is the deep-dished, stuffed variety. There is generally always a wait at the one in the downtown area. There is one on Belmont in the Lakeview neighborhood that is less crowded.
Portillos is great for whatever kind of food. I usually get the bacon cheeseburger, but you can get the chicago-style hot dog there. It is in the downtown area at Clark and I think Ohio. I generally don't pay attention to cross streets.
Museum Campus, which is south of Grant Park, has the Planetarium, Field's Museum, and aquarium. Personally, I'd skip the Field's. The Planetarium was my favorite and I was very surprised by that. But I really enjoyed the hands-on aspect of it. I believe you can buy packages which will probably save you money to visit all the museums (the 3 mentioned, the Art Institute, and the Museum of Science and Industry (which always has a horrific line)). The museum of Contemporary Art is east of Michigan Ave a couple blocks south of the John Hancock tower. Which is also in the area of the huge Hershey store and Ghiradelli chocolate which has delicious ice cream sundaes.
There are many tours: boat architecture tour, ghost tour, gangster tour (I can't remember the exact name of it), and the double-decker bus tour of the city.
For a feel of the neighborhoods, you can take the blue line north-west to Damon and walk around Wicker Park/Bucktown which is considered the artsy/hipster neighborhood. Take the Red Line north to Belmont and experience our pride areas/Boystown. There are many independent stores, like the cookie jar store and The Alley, which is alternative lifestyle central all within a mile or so from each other. Plus, the neighborhoods give you more of a chance to see the architecture in this city, which is gorgeous in my opinion. These neighborhoods also have places like the Weiner Circle, Flub-a-dub-chubs, and other hot dog places plus a variety of other types of restaurants from Mexican to Thai and of course sushi.
Whew, that was probably an over-load of info. I LOVE this city if you can't tell.
Any other questions or clarifications, just ask.