domain name registration is just that... registration of the domain name. As you've just found out, now you have to get your services hosted (web, email, ftp, etc, etc) and DNS setup so that people can actually reach those services.
The easiest (and sometimes cheapest) thing to do is have them all done at the same place, but for a variety of reasons (not being cheapest is one) you may not want to do that.
So to answer your question, YES, you can have your DNS hosted elsewhere. Usually hosting companies will do the DNS as well, so that's probably your best bet and they can guide you through it if you don't know how.
If you are are hosting your own server (in your house for example), then just DNS service is all you need, although I would think your registrar would do that for you without added cost?