Take the Southshore METRA to the Millennium Park Station located at 151 N Michigan. You can then walk 1.3 miles to the Ogilvie Transportation Center at 504 W Madison and get on the Union Pacific METRA which will drop you off at the front gate of Ravinia.
To walk between the 2 stations take Randolph Street West, cross the Chicago River and the first street after the River is Canal. Go south (left) and a block of so down is an entrance to the Ogilvie Transportation Center.
I'm not sure of bus routes, but you can check out www.goroo.com and put in the 2 addresses I gave you and see what comes up on the date and time you will be there.
Rich has given you good directions.
Just my two cents, that sounds like a rather long journey compared with driving, the train ride from Michigan City to Chicago is almost as long as the drive (providing of course you don't get stuck in traffic), then you have to walk across Chicago, another 20 minutes and then the train ride to Ravinia, another 45-50 minutes plus whatever waiting time, by then you've doubled or even tripled your travel time. And then you need to make sure there is a late enough train from Millennium Park to Michigan City
The Chicago Botanic Garden is very close to Ravinia, if you did decide to drive you could make a nice day out of it. There's also a really nice bike trail starting at the Botanic Garden that runs all the way to the north edge of Chicago