If you're parking, the main zoo lot is on N Cannon Drive, just south of Fullerton Ave. See the parking page for more details about parking there or a little information about parking in the neighborhood.
From within Chicago
Most visitors from any distance in Chicago will take Lake Shore Drive from either the north or south, take the Fullerton Parkway exit. Turn west (left if coming from the south, right if coming from the north) onto Fullerton Ave, and then a pretty quick left into the parking lot on N Cannon Drive.
From the south and southwest
If you're coming into the city from the south (on I-90 or I-94) or the southwest (on I-55), you'll follow the same general path. If you start on I-90 or I-94 from the south, take exit 53C to join I-55 North towards Lake Shore Drive. I-55 North will bring you to Lake Shore Drive, and you'll stay to the left in order to go north on Lake Shore Drive. From there, follow the directions from within Chicago.
From the west
From I-290, take the highway into the loop, staying left until just after the exits for I-90/94. Just after those exits, move to the right in order to take the exit for Wacker Drive. You'll stay to the left to go onto Wacker and then the left again to get to Lower Wacker. Continue on Wacker until the very end when it will connect with Lake Shore Drive. Stay in the left lanes to go north on Lake Shore Drive. Once on Lake Shore Drive, follow the directions from within Chicago.
From the northwest
Follow I-90/94 into the city. Take exit 47A for Fullerton Ave, and turn left onto Fullerton, going east towards the zoo. A few blocks after Clark, you'll turn right onto N. Cannon Drive and the zoo parking.