Hi,
Congratulations on getting your first Mac. I'm sure you'll love it.
I'm using an HP Deskjet 5550 and lately I've been having the same problem. In my case, I think it may be connected to the USB cable (just changed it and the problem seems to have disappeared). Try turning the printer off and then on again. If that corrects the problem, I'd suggest changing the cable. HP printers can be fussy with cables and connections so make sure it's connected to a powered hub or directly to the computer and use a good quality cable (like Belkin).
You don't need the driver from HP - Mac OS X 10.4 includes built-in drivers for many printers, including yours (I checked for you). If the cable solution doesn't work, remove the HP driver and remove your printer from the Printer Setup Utility (Applications --> Utilities). Then add the printer and select HP from the drop-down menu next to Print Using in the dialog box that will open after you add your printer. Go down the list until you see HP Deskjet 600/600C - select that profile.
Hopefully one of those two solutions will work. If they don't, post back.
Claire