Chicago offers a variety of ways to get around the city, whether it be by train, bus, taxis, bike rentals, tourist trolleys, and even some water taxi service! Chicago only runs second to New York City in having the largest public transportation system in the United States, so getting around should be the least of your worries during your visit. The city recently adopted the Ventra electronic fare card system, which is something you should consider getting if you're going to be relying on public transit for 2 days or more.
This bike sharing system, Divvy Bike, is new to Chicago and everyone uses it, even the locals. Available 24 hours a day, 7 days a week, you can get a 24-hour pass at any Divvy station for $7 USD and get unlimited 30-minute rides to wherever you want to go (that has a Divvy station). With over a hundred bike stations in the city, including in Andersonville and Lakeview, this is the perfect option for beautiful, leisurely days when you want to run small errands, try different places to eat, or just not feeling like it's a CTA-kind-of day.
You will most definitely hear people refer to "the El" when referencing how you should get somewhere. The Chicago Transit Authority (CTA) has both an underground and an elevated train that operates 24 hours a day, 7 days a week. The train costs $2.25-2.50 USD and allows you to transfer to another train or bus twice within a two-hour grace period. The "Red Line" is definitely the most populated train, connecting you to a majority of neighborhoods in Chicago, including gayborhoods like Lakeview (Boystown), Andersonville and Rogers Park. Expect massive crowds where you're expected to pack in like sardines during the rush hour times of 7-9am and 4-6pm Monday thru Friday.
The CTA also offers a public bus transportation system that includes over 200 bus lines that will take you to literally any Chicago neighborhood you want to visit. Same prices as the train and must be exact change if paying in cash.
Chicago's current taxi rates start around $3.25 USD for the first 1/9th of a mile, with an additional .20 cents USD every additional 1/9th of a mile. There are a handful of companies to choose from, but most people use Flash Cab or Yellow Cab. Flat rates are offered to and from major airports and should be no more than $40 per person. All taxis should also take credit card, so don't let them give you a hard time about that, although it's usually helpful to confirm before the ride starts to avoid conflict. Being totally honest though, most people (with smartphones) usually use the Uber or Lyft apps.