Well.....there is not much to do, really
You have a router that can provide you with Internet connection and does the DHCP......and both computers are connected to the router (I assume), so you should be able to ping your Vista from XP and XP from Vista (unless your firewall is blocking it).
In regards to Windows 2003 Server, I don't know why you think you need it (may be just for learning purpose)....2003 server behaves a bit different then 2000 server, by default 2003 server doesn't allow you to get on Internet or allow incoming connection, you need to enable all that (make sure if your router in providing you the IP address then you have the DHCP disabled on the server).
Since you are behind the Router firewall, disable the firewall on the operating system and run the home network wizard on both computers and also make sure both computers are in the same network group....that should get you started with simple home network for 2 PCs.
Cheers