“Kindle Fire”??? Do you mean a “Fire”? The last Kindle Fire was made some time ago. No one should attempt to purchase a new Kindle Fire, even if they could find one.
Neither does Amazon, since a couple of years ago they dropped the word “Kindle” from its name. Thus, ftg’s comment.
Are you in Wales?
In the US a lot of libraries use the OverDrive service for e-book lending. Here’s the help page for it:
The actual implementation will vary depending on the library but there’s no technical reason why the e-Ink readers should have any problem connecting to this service.
Personally, I think the OP made the right choice with a Paperwhite. It’s got the same screen as the more pricey models and it’s cheap enough that if it gets dropped in the bathtub, it’s not the end of the world. I love books but the convenience of the e-readers can’t be beat.
Ah, it looks as though Overdrive in the States gets on better with Kindles than it does here. 
Here it only wants to talk to Kindles if they are Kindle Fire but it’s happy with all the other e-readers that do epub.
No, I’m in Scotland. I just stole a Welsh word for my username and ran off with it. (I did live for some time in Aberystwyth.)
Most US libraries work fine with the Kindle now (obviously I can’t speak to whether that’s true in the UK). You check out the ebook, then there’s a link to select the format (sometimes; not all books are available for Kindle for whatever reason, though it’s fairly rare), then follow a link that has you log into your Amazon account to get the book added. It will appear the next time your Kindle connects to wireless, and it’s automatically removed after the 3 weeks or whatever.
One interesting thing I’ve found is that if I turn off wi-fi on my Kindle, and the book expires, it is still readable until I reconnect it to wi-fi. Of course that means you can’t get new books until you reconnect.
Depending on where you live, you may be able to join multiple library systems, which greatly increases your book selection. Your existing (well, former, I know this thread is a bit old) library system may still allow you access unless you did something like notifying them you were moving out of their area.
Well, there is - I mean, if you have a tablet you can install the app and read your Kindle books on your Nook tablet or your B&N books on the Nook app on your Amazon Fire. But a non-tablet Kindle - e.g. a Paperwhite - has no way of reading books from another provider (unless the “experimental browser” would allow it; I’ve never tried).
Converting formats is trivial using Calibre; doing so with a DRM’ed book is slightly less trivial, and I won’t go into more detail there because that’s still a gray area legally, but… it should not be a barrier either.
The downside to reading books from another purveyor is that you usually have to download them to your computer and sideload them to the device. It’s a minor inconvenience, and Calibre makes it fairly simple.
The one source of books that IS a problem to buy from, and read on other devices, is Barnes and Noble. They discontinued the ability to easily sideload books a couple years back; you can now read them on a Nook, or through a browser - they got rid of the “download” button so you can’t download their epubs any more. HOWEVER, if you can get the now-discontinued Nook for PC software, that still works and will download the books for you, at which point you can do your thing with Calibre.
Something nobody mentioned, that may or may not be relevant for you: do you want to be able to buy and read magazines? If so, B&N is actually better than Amazon. Amazon deletes issues by default (you have to tell them you want to keep them), AND you can’t reformat them for reading on other devices: I tried using Calibre and it failed. B&N’s magazines can be reformatted for reading on your Kindle.
That is a concept that sounds better than it works out to be in practice. We have a Prime subscription and virtually nothing I want to read is available that way. The Kindle Unlimited is not much better from what I understand (you pay a monthly fee and read anything you want… if it’s available).
No, a Kindle can read books from other sources just fine, provided they’re in a format the Kindle recognizes (or you convert them to such a format) and aren’t restricted by that source’s DRM. (This does rule out ebooks from Barnes & Noble, unless you convert them yourself first, but not from other sources, like for example Project Gutenberg.) Books and other documents can be transferred to the Kindle via USB cord, (Send to Kindle - Email), or computer app.
Mama Zappa:
I actually use a Nook Simple Touch myself. Pretty old in the tooth (2011) by ereader standards. Between using an older model and rooting and romming* the ST right away I’m not up-to-date with the current B&N ecosystem. E.g., I use FBReader on it. It was true at the time that one of the apps you could load to a Nook was the Kindle reader software. Seems like things changed.
From what I read when I last shopped for an ereader (Mrs. FtG’s Kindle) the Nook world didn’t seem like an option for us.
(I also got a Nook HD+ tablet. Really good, large-ish tablet except heavy. I use a Fire now for day-to-day use. Not as good a screen but easier to tote.)
- Sadly, the Android ROM universe is in trouble. CyanogenMod imploded and the replacement group LineageOS isn’t up to speed yet. Making romming many things a task that requires even more expertise than before.
Being able to root (at least) a device eliminates a lot of lock-in issues. But that is generally getting harder and harder to do. Shame.
True - I didn’t specify “no way to read books from elsewhere without fixing them first”. You do have to use some tool to convert them to a Kindle-able format.
But “out of the box”, you can’t read an ebook from B&N (or Kobo’s bookstore, I assume) on the plain Kindles. There’s no app to permit it like there is on one of the tablet-style readers such as the Fire.
Converting a non-DRM’ed book’s format is trivial using Calibre, at which point yes, you can sideload or email it.
Update- bought a Paperwhite, worried for about one minute how I would get enough content, then went to the library. Libraries, that is, as I have cards for 3 big city libraries and will currently get at least one more. Websites, that is. There are enough books, that I want to read, to last me years and years. Won’t ever have to buy a book. And, I can return them with a click. I stopped going to the library in the first place because I would take too many out, not return them in time and rack up gigantic fees. No more of those shenanigans. Holy Swiss Army knife! I LOVE MAH FRIGGEN KINDLE AAAAAHHHHHRRRRGGGGHHHHH cough cough. As you were.
Woohoo!! Welcome to the club!
BookBub sends out daily emails listing free or reduced price books based on preferences you set up, e.g. love cookbooks, hate adventure stories or whatever. The freebies are rarely anything special but sometimes you can discover a new author you’ll enjoy.
If you wanna really treat yourself, get a cover from Oberon Design. My first e-reader was a Nook Color, and my husband had given me one of their cases… and the thing was stolen 4 months later. I regretted the case far more than the Nook - but any replacement device I might have purchased wouldn’t have fit the case so I was sorta unable to upgrade :D.
With the Paperwhite, as Amazon releases new versions, the size stays close enough that your older one’s case will still work (so, having replaced the Nook with a Paperwhite, and having received a new Oberon cover for it, I can upgrade my Paperwhite to the latest once the budget allows :D). I actually have the Roof of Heaven design that shows up when you first click that link, and in the same color.
Library books: It’s sort of annoying that with Overdrive books, you can’t return them directly from the Kindle. I mean, they’ll time out after 3 weeks but I’d like the ability to easily return them if I’m done early, to free them up for someone else (the library only has a license for so many digital copies, so there CAN be wait lists).
If you like science fiction, try Baen for books - they have a lot of big names, and their books are DRM free and available in all formats (no need to convert). Supposedly their books sold through Amazon and B&N are also DRM-free, but it’s nearly impossible to tell for sure. Weightless Books also sells DRM-free stuff but its selection isn’t as good - sort of niche SF / fantasy / horror, though they are the purveyor for Clarkesworld magazine and one or two others.