Just in case if your new computer is a new Dell 4700 or 8400, be aware that it no longer provides 4 IDE ports.
The new motherboads with PCI Express include 2x IDE [PATA] ports (1 channel, master/slave) plus 2x SATA ports (1 channel, 2 drives).
So then if your new hard drive is a SATA drive, you can only put 2 IDE [PATA] drives in your system and not 3.
You can of course purchase PCI based IDE controller cards (sometimes for virtually nothing when people got extra ones included with "big" hard drives of the past in order to use them) to extend the amount of IDE you can have.
As far as DVD->DVDR---I agree with the others, copy to hard drive first.. more reliable than on-the-fly. Plus I would really suggest, IF you have room/power/connectors, keep both, and use the DVD player just to play DVD's and save the laser on your burner. Though it sounds like you're quite full as it is.