Chicago has excellent public transit, and Shedd is in a great location to take advantage of it. The bus will drop you right at Shedd's door, or you can take the L from within Chicago or regional commuter trains from the south to get you pretty close. Google Maps can help you piece together public transit routes to get you from any location in Chicago (just choose the public transit option, which should be already selected if you use the previous link).
To walk, go east on Roosevelt towards Grant Park. Once you cross Columbus Ave, veer slightly to the left to go into Grant Park, but staying mostly parallel with Roosevelt. That path will take you through a couple of tunnels and you'll see the back of the Field Museum on your right. Keep going straight ahead past the Field Museum to get to Shedd Aquarium.
The South Shore Line goes south and into Indiana, including Michigan City and South Bend.
Both the Metra Electric District and South Shore Line bring passengers to the Museum Campus/11th St. stop. From there, follow signs east to the museum campus. Once you go under a bridge, you'll see the Field Museum on your right; continue straight to get to Shedd.
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