You have a choice really...you could set up the drive so it is separate from your internal, like having a hard drive on D: and E: in windows; they'd probably show up as /dev/hda and /dev/sda. (I'll make that assumption here).
As for copying everything over to your new drive seamlessly; that should be fairly trivial. It'd probably be best if you booted from a third drive, like your rescue disk. Reformat your new drive to have a Linux filesystem:
Then mount both the drives:
mount /dev/hda /drive1
mount /dev/sda /drive2
And copy from 1 to 2:
cp --archive -v /drive1/* /drive2
It's also possible to combine the capacities of both drives into one big volume; this is slightly more complicated, but either Linux software RAID or Linux LVM can be used to achieve this effect:
Linux Software RAID Howto
(You'd want to use Linear mode, or maybe Stripe mode if both drives are the same size).
Linux LVM Howto
(This is generally more flexible, and probably more desireable, though I don't have any personal experience with it)
Either way, though, if you did this, your external drive would be a pretty much permanent fixture; it wouldn't work on other computers without being reformatted first, and your computer's internal hard drive wouldn't work without it, since all the data will be spread out between the two.