After you get it updated and registered, you should be able to find free books on Amazon, both out-of-copyright classics and newer promotional offers. I go to the "top 100 books" section, and switch to the "free" list (default is "paid"). There is lots of iffy stuff, but occasional gems offered for a short time, so I check often. If you narrow your search to a favorite genre, you can see further down the list - e.g., the historical fiction list is a subset of the entire list, so the top 100 there may reach down to #850 on the full list. Keep searching different genres to see more, and exclude those you are not interested in.
As far as library ebooks, my library has multiple access options. Overdrive generally has one, maybe 2 copies of a given title, so you have to wait for them, but they are downloaded to your device. Hoopla has unlimited copies, so no waiting, but leans much more heavily to online use. You can download some media to mobile devices, so look more deeply. There is also One Click Digital, which looks like out-of-copyright classics, primarily. And, as others mentioned, Gutenberg Press.
If the kindle you have is an older e-ink device (not a Fire-type tablet with internet capability) accessing anything but Amazon involves checkout thru your PC and transfer to the kindle. Not hard, but you just can't quite do it directly from the kindle.