Buy the domain itself wherever you want. I like cloudflare, and a lot of people also suggest porkbun.com. You then point the nameservers for your domain to whatever DNS service you want. If you stick to cloudflare then it's already done for you.

For dynamic DNS I use cloudflare's one using my router to keep it updated. It's easy to set up. Depending on your router you may need to run a service on a machine to do this instead. things like pfsense/opnsense should have it built-in.