I think you will need the DNS addresses - try having a look on your ISP's website for their tech help, or check any documentation you received when you signed up.
As for lack of success with ping - this could be more of a problem. Check to see if XP has enabled its firewall (look at the internet connection, see if it says it is firewalled); it's just possible that the Windows firewall isn't allowing pings, but my hunch is that this isn't the case. Otherwise, double check that you've fitted a crossover cable between the machines, not an ordinary non-crossover type (it's caught me out before). When pinging, are you using the numeric IP address, or the name of each machine?
Resolv.conf - you'll need to be root to edit this file. Easiest (quickest) way is to open a terminal window (you're using Red HAt 8, so I'm assuming you're running Gnome - look in the menu for 'terminal' or similar). Then change to the root user by typing
Enter your root password (you won't get any feedback on screen). When successful, you can edit the /etc/resolv.conf file by typing
This will open the file for editing. Press the 'insert' key on your keyboard, then type in the names of your nameservers, as outlined previously. When you're done, press the 'ESC' key, then type : (colon) followed by 'w' (omit quotes!) to write the file. To get out of Vi, type : then 'q'.