Chicago Botanic Garden - Best Way to Go
 

not sure if you are into Cycling but the botanic garden is a fantastic place to do it.
Glencoe will be the closest metra. I think it is walking dist.

if you wanted I think you can bring a bike on the metra on weekends.

Plusses and minuses of all suggestions:
Davis - not a good choice; this is the stop in northern Evanston, at least a 40 minute bus ride away.
Glencoe stop, then Pace bus - this should be a pretty quick ride, maybe 15 minutes, from the train station.
Braeside - this is a tiny stop not served by bus lines. It's about a 1.5 mile walk to the entrance of the garden through residential areas, nothing like the sprawling Lake-Cook corridor west of the expressway.

Remember, once inside, you have a lot more walking to do. Do see the new Evening Island and skip the prairie walk.

Bikes will get you from the train to the garden, but can't be taken inside the garden. Bring a lock.

There is a continuous sidewalk from Botanic Garden to the Braeside stop (corner of Lake-Cook and Green Bay Rd). It runs on the north side of the street. It would be a 20-30 minute walk depending your pace.

Unfortunately, Metra no longer allows bikes on trains. After a highly restrictive pilot program in 2002/2003, they decided it was not worth it to offer the service. It was so restrictive and hush hush, nobody knew it was being offered or the proper procedures to do it. But, that's a whole 'nother rant.

To expand the bike idea, you CAN bring bikes on the CTA on all weekends and non-rush trains during the week. You could ride the red line and purple line to Linden Avenue (end of the line) and then ride through some residential areas to the Green Bay Trail which starts at Lake and Green Bay Rd. Then ride north 7 miles to Lake Cook Rd. and west on the sidewalk mentioned above. Sounds complicated but it's not really. You can bike through the grounds (north/south) on the east path only, which is separated from most of the garden areas. The CBG is the northern most point of the North Branch bike trail which runs south along the forest preserves all the way to Peterson Avenue and Billy Caldwell Golf Course in Chicago.

Chicgal, wear VERY comfortable shoes when you go to the Botanic Garden! It is huge, and so beautiful you will want to see ALL of it. Rest up and drink lots of water because it will be hot. And please smell the roses for us!

I don't know the trains in the area and can't help you get there, but sometime take the time to visit it during the Xmas season. Their Xmas show is fabulous. I also drive 3 hours just to go to the Antique and garden fair in April. You also sound so cute and young. I've got a single (cute) son in Lakeview. I've decorated his condo now I'm looking for the mother of my Grandchildren (to be)!

Sound lovely, Chicgal! DH and I enjoy our trips to the Botanical Gardens in NYC - only hope I can convince my sons to visit the gardens with us while in Chicago ;)