I use a free service for my DNS registrations. In my case www.zoneedit.com
, but there are others as well. The process is pretty simple - set the DNS up on the nameservers (it's pretty simple to understand their instructions), and tell your domain registry what nameservers to use (these will be given to you by the DNS provider).
Depending on how your network is set up will determine what needs a static IP. For example, my router is a combined router/modem that has its IP set to the static IP, and I simply forward the relevant ports (in the case of web access 80 and 443) to the relevant machine on my private network.