Getting a domain is easy, just go to Namecheap, Godaddy, or any number of other registrars, and buy one. Actually doing something with it can be much more difficult.
By far the simplest thing is to use some service that will host email for you. Some only allow a very limited number of addresses, others it is unlimited. Expect to pay for this service, but it shouldn’t be too much. I think Apple with iCloud can do it as low as $0.99/month.
I host my own on a server I pay for “in the cloud”. Which really means I’m renting some space on someone else’s computer. Even for people who know what they’re doing, I strongly discourage this. Running an email server is one of the most annoying and tedious things you can do.
There are lots of good reasons to rent a virtual server, but hosting your own email is not one of them. When I started, many years ago, hosting my own email was the only cost effective option, and I only continue to do it through stubbornness and inertia.
I didn’t run my own email server, but I do have my own domain and ever since the early days had used a mainstream email supplier aimed at SMEs. With the advent of the huge webmail services like yahoo, gmail, msn/live/outlook.com, etc., my supplier, although still growing, was relegated to the ever more minor league fringes of the SMTP ecosystem.
As the spam–antispam wars hotted up and SPF, DCIM, and other band-aids became first necessary and then insufficient, I ended up with several major ISPs that would not deliver my mail. My supplier had enough bad actors among their 10s of thousands of customers that it was easier for e.g. gmail to simply block all mail from my supplier’s IP addresses. Mostly intermittently, but that was enough to render email way too unreliable for my business use. That sucked.
Moved my mail service to outlook.com and re-pointed my MX, etc., records there per their guidance and now my mail gets through to everywhere.
We’re definitely past the time where roll-your-own is the right answer for email interoperability in the face of the malware / spam onslaught.