I can't help you with the online part because I don't even have the network adapter but for burning, you should get DVD Decrypter (it's a freeware so try searching it with google, it should in the first page). So basicly all you have to do with that program is enter mode iso read to make save an image (iso format) on your hard drive then you go to iso write and burn that image. It's really that simple. For the media, it's a little complex, it depends of your PS2 version (http://www.mod-chip.com/ps2version.htm), older versions (V1-3) can't read DVD+R and the others can (mine is a V4 and it reads DVD+R perfectly) but all versions can read DVD-R, I know people are using the following brands: TDK, Maxel, Verbatim, Ritek (AKA Ridata) and Memorex. I personally tried Verbatim and Memorex and everything worked perfectly but I know some people have problems with Memorex discs, I will soon try HP discs.
That's the best way to make a copy but I suggest you use DVD Decrypter it's really simple and fast. Make sure you don't make a copy on the fly if you use Nero and it should work.