The Name's The Thing
You can set up a website without a domain name. Most hosting companies have a free tier, so you can set up a web presence for nothing. But I don't recommend that.
For much the same reason as I don't recommend relying on a social media platform, if you have your author presence at 'fancypants.blogspot.com' what do you do if you don't want to use Blogspot any more?
Buy a domain name. They're not too expensive, easy enough to connect to your web host, and you can keep them for life. You can change whoever hosts your website at the push of a button, and your visitors never have to know.
I recommend buying your domain name separately to your hosting. Most hosting companies will offer you a free or cheap domain for the first year, which admittedly will often be cheaper. But after that first year, they can charge whatever they like for a renewal. You can technically move your domain to another registrar, of course, but that's a hassle, and runs the risk of your site being down for a while during the transition.
Keep them separate, and you keep the control.
A lot of folks go for the classic 'firstnamelastnameauthor.com' (including me, markhoodauthor.com) .
If you can get just your first name and last name, you might also want to go for that - although if your name is particularly tricky to spell, it might be worth picking something else. Author Caimh McDonnell (pronounced Qweev) went for 'whitehairedirishman.com', for example.
Another option is to take something from one of your books, perhaps the setting or title of one - although be careful this doesn't box you into a corner. As your career develops, people are more likely to want to follow you, than one book series.
If you write cross-genre, it might be worth considering a site for each genre. If you have more than one pen name, I'd say you definitely want a site for each.
Don't agonise over this - you can always add more domains later, if you diversify your writing. Or you can 'park' an old domain and send it to your main one. All my writing is at 'markhoodauthor.com', but I have a couple of others that just redirect there, rather than having their own website set up.
And I would advise getting a .com, rather than any of the 'new' top-level domains. The others can be cheaper (though not always), but are mostly aimed at certain industries (.bar, .accountant, etc.) Maybe you're a horror author, and want .boo as a top-level domain - by all means, go for it. But consider whether it gives the impression you want it to. Also check renewal rates, sometimes they hook you in with a cheap first year and then ramp it up.
And finally you might find some readers still want to add '.com' to the end anyway, and you end up explaining it every time.
At the end of the day, though, most people click links or search for something, rather than typing in a whole URL. It doesn't matter as much as you might have been told.
There are almost as many domain registrars as there are domains. Picking one can be tricky.
If you want to get your domain through your host, that's fine. I advise doing it separately, though, for the reasons I already explained.
So who do you pick?
There are a few that usually come up when you ask for advice, and I've listed the most common below. Being on this list can't be a guarantee of perfect support, any more than absence from the list meaning they're not legit. These are some I either have personal experience of, or recommendations from people I trust.
NameCheap.com
My personal recommendation. I've used them for years, their prices are fair, and they've never given me any trouble.
Domain.com
I've used them in the past, they tend to do well in customer surveys.
GoDaddy.com
My first website had a domain from them, but they have a poor reputation. Anecdotally I've heard about poor customer service, domains become more expensive after you search for them, and they can even auction off domains sometimes.
Whoever you go with, ensure they're offering 'Domain Privacy' - most do, at no extra cost. It means that your email and home address don't get shared with the world when you register.
I'm going with NameCheap.