NO HP drives have BURN-Proof since they are just rebadged Sony drives and Sony is to cheap (arrogant) to pay Sanyo for a BURN-Proof license.
As far as the software goes, if you are new to CD burning then I would use Easy CD Creator, if it comes with the drive you buy. EZ CD Creator is pretty foolproof but Nero is much better for most experienced users. After you become familiar with burning you can try different software packages until you find one that fits your needs. Nero is very good, I own a copy but barely use it.
I like...CDRWin for most backups, CloneCD for those hard to do backups, and Feurio for music (this ones hard to beat IMHO). I use Nero for making data CDs but thaats about it.