As an aside, I would recommend against using Fedora for a server. The lifecycle is so short, that you'll either have to rebuild it in a year, or you'll stop getting security updates (which is particularly bad for a web accessible server). If you like/are familiar with Fedora, look into CentOS, the free version of RedHat Enterprise Linux that has the same feel, etc as Fedora.
If you don't want, or can't get a static IP, you can use a service like dyndns (see www.dyndns.com
for example) and this can be automatically updated from you system as yoru IP changes. It is preferable to have a static IP though.
In terms of registration, assuming you can find a registry easily enough, I think the cheapest option (and it works fine) is to register a domain name and use one of the free domain name services that are around (I use www.zoneedit.com
and there are others around) to point to your server. Zonedit allows up to 5 domain names for free, so that shouldn't bee too much of an issue. You will need to register the namservers with the registry, and from there set your server up.
In short - register your domain name and pay the relevant fee, set up nameserving and set up you server.
Hope that helps