How do I get to major tourist sites from Union Station?

Check out the RTA's online trip planner:

http://tripsweb.rtachicago.com/

Also go to the CTA's webpage for information, instructions and route maps. Here is the URL: http://www.transitchicago.com/

You can take the Red Line L train south to Chinatown and get off at 22nd/Cermak. From Union Station, exit the west side and walk south 2-3 blocks on Clinton to Blue Line L stop (under the Congress/I-290 overpass) and board train toward O'Hare. Go three stops to Jackson and transfer to the Red Line train towards Dan Ryan/95th. Cermak/Chinatown will be 3rd stop to the south. Follow your nose to Chinatown from there--it's right there. Exit the L to the WEST-avoid going east of the tracks--trust me on this.

For Mag Mile, Navy Pier and the Museum Campus (Field Museum, Adler Planetarium, Shedd Aquarium & Soldier Field) you are within about 2-2.5 miles from all of them at Union Station. It's not a bad walk on a nice day. You can exit Union Station towards the East, cross the river and look up. The Sears Tower will be right there. Take a left on Wacker Drive as it snakes alongside the Chicago River. Once you cross Lake Street (Green Line L tracks overhead) Wacker will curve right (East) and you'll see some of Chicago's best architecture. Go East along Wacker and you'll hit the Mag Mile where the river and Michigan Avenue intersect. Wrigley Building and Tribune Tower are the first buildings you'll see. Follow that north. To get to Navy Pier, take Michigan Avenue north just a couple blocks to Grand Ave--just north of Tribune Tower on your right. Head down the stairs to Grand Avenue and walk East until you are at Navy Pier.

A cab ride is the quickest to any of these places in my opinion--you might end up spending $10 max per leg.

If you want to keep on a budget and still get around efficiently, CTA (includes city buses and the L train) is the best. Get yourself a CTA Visitor pass for $5/day and you'll be good to go:

http://www.transitchicago.com/maps/fares.html#e

Have a blast!