Babies - Chicago Children's Museum (and Navy Pier)

Kids Town and Treehouse Trails both have areas specifically designed for non-walkers. Kids Town's area has a low wall around it which makes it easier to keep crawlers contained, although big kids tend to jump in, too. Telling them "this is just for babies" usually does the trick. Treehouse Trails' area is more out of the way so you're less likely to have issues with big kids, but it doesn't have any boundary to keep crawlers in.

The Pritzker Playspace is a great area to play in, though it only allows preschoolers and younger so it may not work if you're visiting by yourself with bigger kids as well. Hours are very limited, so check that page for more information.

Babies will also enjoy splashing in the Water Ways, although there's nothing down low so you'll need to hold them up the whole time.

Dinosaur Expedition has lots of small pieces of rubber, so avoid it for ages when everything goes into their mouths.

Nursing

Chicago Children’s Museum explicitly encourages nursing anywhere in the museum that you feel comfortable. Kids Town has a special Caregiver Center, where you can have more privacy. Outside of that, Kids Town and Treehouse Trails have chairs specifically designed with high arm rests to make nursing easier. These chairs are nicely located within sight of the exits to the rooms, so you can feel more comfortable letting older kids play and make sure they don’t leave the room.

Navy Pier also has two locked nursing stations, almost like pods, one of which is almost directly across from the Children’s Museum (the other is a very long walk away). The nursing stations are not large, so it’s not great if you want to go in with older kids, but they do offer complete privacy. You can go to the information desk by the front entrance) to get an access code.

Bottles

Kids Town and Treehouse Trails have seats designed with high arm rests to make giving bottles easier. They are nicely located, within sight of the exits to the rooms, to make it easier to keep an eye on older kids and making sure they don't leave the room.