Can i transfer data to other computers by swapping out internal hdd

Hello, Got limited supplies, 2 sata cables, drive with os and drive i just got new. I installed it and the computer need me to partition it before i could access it. I want to transfer important info to the new drive, reformat and reinstall os on old drive, and then move all my important stuff back onto the original OS supporting drive. That final part is important because like i said, only 2 sata cables and one will be swapped out a few times to operate my cd/dvd-rw drive to install os and then transfer data from hdd's and then ultimatel ask the CD/DVD drive again. Sorry this is sloppy and poorly written Im in a hurry (to type this) and my desktop is crammed with stuff , half the keyboard is in the tower and some off the desk.