Relays do cost money, though I think some have a free tier for small volumes of mail. You might also see if your registrar or host provides anything for email.
The easiest way to do this is to start with just receiving email and not worrying about forwarding, though. You can host your own imap server and just have a catchall account that’s separate from your main email to start, and if you really want to forward you can worry about send later. Receiving email is easy, the thing that people struggle with for email is sending because there are a few requirements like dkim / spf / DMARC and reverse dns that you might not know about and may configure incorrectly and feedback is hard. Also if you have a residential ip I’ve heard it can be harder to send too. If you’re just forwarding to yourself, though, that’s probably a little easier because you can test more easily / mark yourself not spam. If this is your use case I wouldn’t worry about setting up a paid relay service. You don’t need it unless you really want to forward and have troubles making send work in your own.
With all that said maybe anon addy is easy to set up on your own and gives you what you want. I wouldn’t know! I’ve never used it before.