As a librarian, I have to say, we have books and programming/events for all ages, including "Baby Bounce," which is a lap storytime for babies. Toddler storytime is for ages 18-36 months, but some libraries do younger age ranges. And most libraries these days aren't totally silent affairs--I'd honestly be surprised if you found a children's area that's totally silent. ;D Most families bring in all their kids--from infants to school ages to teens. We have stuff for all of them.
Libraries often have "board books," or those cardboard extra sturdy titles that kids can use as toys, as well as developing print awareness. Kids should be read to regularly from a very young age to develop early literacy skills. Some of our branch libraries do a program called 1,000 Books Before Kindergarten, which is a challenge to parents to read a lot before their kid starts school. You may end up reading the same book 100x (kids love repetition and it's good for them!), but it's excellent preparation for school. Parents are a child's first teacher.
The most important things you can do to develop early literacy skills with your child: talk, read, write, sing, and play. Yes, play is actually very important work for kids. Some libraries now have "smart play spots" that are designed for kids to do imaginative play. In one of our branches, we have a store, a fishing pond, a puppet theatre, a giant alphabet board, and more--kids love this area.
One of my favorite things to do with kids is called dialogic reading--asking them questions about the book we are reading together, and not just "yes/no" questions, but rather, "What do you notice on this page? What do you think is going to happen?" type questions. Or I point out interesting things--"there's a doggie! you've seen a doggie at grandma's house!" to make the connection between stories and real life.