I've used my own domain for 9 years with paid hosting thru a major host. Personally I can't stand webmail and stick to traditional POP3 email and for that purpose it suits me. But it is easy enough to set up domain forwarding to services like gmail if you choose (most likely for a fee).
The nice thing about buying a domain is you can pretty much set up unlimited email addresses under the domain for any purpose you choose, or use a single email address as a "catch-all" for said domain. Web services like Facebook won't know and won't care.
As for specific hosting recommendations, they are all about the same in terms of terrible service and support, but I encourage you to research and decide for yourself.