Sorry guys I am a newbie at this. If I register a new domain name, like with eNom, and I open a new account with a web host, is it up to me to go in and point the new name to the new host? If so, how long does it take before my new site can be accessed on the web?

Yes, it would be up to you. Your host will give you nameservers (ns1.yourhost.com, ns2.yourhost.com) and you log into your domain at your registrar and enter those in the fields. Takes about 2 seconds to do, and then you should expect 24-48 hours of time for propogation (sometimes a bit quicker, sometimes a bit slower).