Any of the latest burners from Pioneer, NEC, BEN Q, Sony, or Plextor will serve you well. I don't copy games, but I do back up DVDs. Programs I recommend:
DVD Decrypter (removes protections, rips, and burns) FREE
DVD Shrink (removes protections, compresses to fit DVD5) FREE
DVD FAB Decrypter (removes protections, rips) FREE
DVD43FREE (removes protections on-the-fly) FREE
AnyDVD (removes protections on-the-fly) $30
Nero Recode2 (rips with DVD43FREE or AnyDVD, compresses to fit DVD5, burns with BurningROM) $8 (Nero OEM Suite 3)
Intervideo DVD Copy 3 (rips with DVD43FREE or AnyDVD, compresses to fit DVD5, burns) $50