We have a very nice parish library. We have a volunteer committee (I’m a member) headed up by a Librarian (she volunteers for this). She taught us how to catalogue books and we all spent time doing this. All of our books are catalogued, and they are checked out like in a regular library. It took a while to set it up initially, but it’s worth the effort. Recently, though, we were given money from an endowment and got a new computer and an autolibrarian system, so now the books are catalogued in the computer. I’m not that familiar with it so far. Anyway, that is not necessary, just convenient. If there is a librarian in your parish, maybe he or she would volunteer to help set it up and teach how to catalogue.
We meet 3 or 4 times a year for library business and order books once or twice a year through a local Catholic bookstore at discount rates. We also take donations, but there are stipulations to the donations, because we found people were dropping off any books they wanted to get rid of and many of them were musty and moldy or not suitable for a library in a Church. We also do AV material.
The library is open on the weekends between Masses, staffed by volunteers, but people also can go in during the week and check out a book. We have a poster that tells them how to do this.
The library is not as well-used as we would like it to be, so we have a promotional raffle, which has increased business somewhat. If you check out a book, you sign a raffle ticket, and then we have a drawing at the end of the month for a $10.00 gift certificate from a local bookstore. We don’t do this every month, though, and usually someone from the parish donates the money, mostly one of the library volunteers.
The initial setup is a lot of work, many volunteer hours were put into it, so be prepared to put a lot of time into it. And if you start out by asking for donations, that is a good way to get started, but be sure you have published the stipulations in your bulletin. We do have a budget to work with, thankfully, otherwise you have to find a way to raise some money to buy books. But it is a lot of fun, and a good way to meet other parishoners.
Good luck in your efforts.