My old Kindle gets its content wirelessly and not from our home internet so that e-reader is pretty easy to get content for.
I fear you need to discuss more with the library how they get content to what reader (it matters.)
I saw an article in a local paper that the large city library is now "lending" books electronically. Since my local library' selection is awful, I'd love to have an e-reader. BUT. We have dialup internet service. I just did a bandwidth test-- 20.4kbps. Molasses.
Would it take forever to download the library books (or any book, for that matter)? Is it just not worth the effort? I do not know much about e-readers, so I will heed your advice...