In elementary school, you couldn't go to the library on your own. You had a weekly library class, with the day of the week depending on your grade level. That was true in the two different elementary schools I went to; one was the usual K thru 6, where I went up to 4th grade. The other was K thru 8, where I finished the rest of the grades.
At the first school, grades 2 thru 6 had one designated library day per week for each grade level, where students could go to the library during their recess, but only on that day. Kindergarten had a classroom library, and 1st grade was taken there by teachers during some reading classes. I didn't end up using the library very much, because I was paranoid of losing a book (after I lost one of my textbooks).
At the second school, there was a weekly library class, which involved a lesson on things like research, critical reading, etc., plus time to browse. Students were barred from going inside outside their library class. The room was also used for rehearsing for the graduation ceremony. I didn't check out books very often, because I was embarrassed of my classmates seeing me reading "dorky" (read: non-fiction) books.
In high school, the library went by the name "study resource center", and you could go there anytime you pleased, as long as you didn't have a scheduled class. The school had a fairly liberal internet policy: any appropriate site other than personal e-mail or games was allowed; even fantasy sports were tolerated. I spent quite a bit of time using the computers.