Ok, having had experience with a fair string of libraries for ebooks and paper, your first task is to examine the libraries online.
1. Find out if they require residence to get a library card (because your library card is your username and login).
2. Find out the size of their collections. Assuming, for example, that you are a reader of mysteries, just go to the catalog online and do a search for all mysteries in the collection. If they've got ~200 of them, it's just a tiny number. If they have 16,000 of them, there's your library!
Many of the libraries have Overdrive installed on their server. That's just to handle the process of borrowing and searching; it does not mean that you have to have Overdrive on your Kindle. (If you are not reading with a Kindle, however, then you may need to install Overdrive on your device.)
I find it easiest to combine reading on the regular reading Kindle with searching for books, reserving, making requests, etc. best done on my laptop. The regular reading Kindle is just awful for the purpose of use online, nor does AMazon try to make it into an internet device; but as a reading device, it reigns supreme.
:)
Lynn