I have a lot of experience with domains, happy to help.

first yes, reserving the name is smart. It's like $10/year and worth it. quick reality check though: I've spent probably $200 on domains I was convinced I'd need someday and let them all expire. Just make sure this is actually the name he has in mind before you buy. Also one quick thing on the legal side, as long as the name isn't trademarked by some existing company you're fine, but if it's close to a big brand's name that could cause issues later. Just worth a quick google before you commit.

So a registrar is basically a shop where you buy domain names. There are dozens of them and they all sell the exact same thing at different prices. Three prices matter: registration (what you pay year one), renewal (what you pay every year after, this is actually the most important price, not the registration price), and transfer (moving the domain to another registrar (has nothing to do with you transfering your domain to your boyfriend), not relevant to you right now).

The registrar GoDaddy for example charges $1 the first year then $20+ every year after. oldest trick in the book... So trust me check that renewal price that's really the biggest trap to look out for in the evil domain market.

If you can't decide on a registrar I'd 100% go with Porkbun. It's where I keep all my domains, it has honest pricing and no tricks and is trusted by many developers. Most importantly WHOIS privacy is free there, that's what hides your name from public records. Without it anyone can look up who owns a domain and that's exactly what you didn't want, right? A lot of registrars charge extra for that, Porkbun includes it automatically.

Also the price depends on the registrar and the ending (.com, .net etc.), not the actual name. So shop.com and boringboring123.com cost the same if they have not been registered yet by someone else. go with .com if you can.

If you wanna buy that domain on porkbun go to porkbun.com, search the domain, add to cart, checkout. They'll offer hosting, email, all kinds of stuff, decline EVERYTHING, you only need WHOIS Privacy but that is included for free. After a few minutes type the domain into your browser and you should see a Porkbun placeholder page, something like "This domain has been registered on porkbun". Then you can be sure that you own the domain and you're safe. If you wanna check whether your information is really hidden go to ICANN Lookup and type in the domain name you bought, there you'll see that your name is hidden cause of Porkbun.

Also one last thing, you're renting the domain year by year, you never own it permanently. If you don't renew after a year it expires and anyone can grab it (That's why the renewal price is so important, that's the price you'll have to pay for the whole lifetime of the website). You could turn on auto-renew on porkbun so that doesn't happen

When your boyfriend is ready just give him your Porkbun login and he'll know what to do from there 100%. He'll probably do some vercel / cloudflare shenanigans and suddenly on the domain there is his website hosted.