If I understand correctly, you do have a computer working for you now, but you have some older drives with valuable data files on them you can't access. The easiest way to work on the older drives will be to install each one in turn in the working computer and try a few tools on them to diagnose. For now, let's work on identifying what you have so we can advise how to set up and connect and make some preparations. When you can feed us back some info we can give you explicit instructions on how to test the drives and recover what you can.
First, though, we need to know what type of drives are the old ones, and what machine and Operating System you had them in. The two most common types are IDE (or PATA), and SATA. On the back edge of an IDE drive there is a power input connector on the right with four recessed pins in a line; to the left are a few pairs of pins for jumpers; to the left of that is a 2" wide connector with 40 pins arranged 2 rows of 20, but one pin is missing. On the back edge of a SATA unit there may not be any pin pairs for jumpers, there will be one wider connector (1½") with 15 fine pins for power input, and another half as wide with 7 pins for the data cable.
If you can tell us exactly what the label says on each HDD - manufacturer and model number - we'll be sure what drive you are dealing with.
What can you tell us about what Operating System (version of Windows??) was on the old machine these drives were in?
Briefly, the way to install each type is a little different, but not much. In each case you have to connect a power supply from wires that come from the PSU, and a data cable between the HDD and an appropriate mobo port. For IDE units you also have to set jumpers on the back edge pins to establish a Master or Slave identity, and that comes in conjunction with what is already installed in your working machine. (There is no such adjustment for SATA drives.) So for that purpose we also need to know what types of drives you have in the working machine, and what ports and cables they are attached to. Once those details are set and the drive fastened in mechanically, there MAY be a few things to set in the BIOS Setup screens, but it may all just work without that.
Now, in preparation for what to do once the connections are made, you will need to download some utility software. IF your old drives are by Seagate, go to their website and download their diagnostic package, Seatools for DOS. There are at least four version of this, so get the right one:
(a) Seatools for Windows runs as a Windows application, but it does slightly fewer things and must have Windows running if you want to use it. That's why I prefer the "for DOS" versions.
(b) Seatools for DOS (floppy disk) is a compressed .zip file of what needs to be put on a FLOPPY disk to use. If you do NOT have a floppy disk drive and diskettes, ignore this one.
(c) Seatools for DOS (Optical disk) is the most commonly used. It comes in two forms. One is an .iso file, which means it is an exact image of all the stuff that needs to be "burned" onto a CD-R disk to use. To do that job you need a CD burner drive and a CD-R blank disk, and some software that is able to "burn" an .iso image to the disk. Nero is one example of software with this ability, but other optical disk burning software does it, too. You have to put the blank disk in your drive, start the software, and tell it to burn onto it the .iso image you have downloaded.
(d) Seatools for DOS (optical disk) again - the other form is a .zip file which actually contains the .iso image file in compressed form so it is just smaller and faster to download. If you get this one, you must un-zip it to create the .iso image file, then process as above.
What all that does is allow you to make your own CD-R copy of the Seatools for DSO utility package used to diagnose HDD's in your machine.
IF your old HDD's are from Western Digital instead, go to their website and get their utility "WD Data Lifeguard". It comes in those several forms and is handled the same way. It just is custom suited to WD hard drive units.
IF your hard drives are from another manufacturer, either the Seatools or WD package can do some of their testing on other makers' drives anyway, but just not everything. If the real maker of your old HDD's has a similar diagnostic tool set available on their website, get that and prepare the required test disk.
In all these cases to use the diagnostics you must ensure that your BIOS is set to boot from your optical disk drive, put the test disk in it, and boot. The machine will boot from that CD-R and load a mini-DOS Operating System into your RAM, and you can use that to run all the tests from a menu system. This works even if you have no properly functioning HDD in your machine because it does NOT need Windows to load first.
So, gather the information I suggested, and gather and prepare a diagnostic tool disk in preparation. As a look ahead, when we get your disk(s) attached I'm going to recommend three stages:
1. We'll use a couple of Windows' built-in tools to examine each disk for a few very simple things that might be easy to fix right away.
2. We'll use the diagnostic test disks to see if there are actually any hardware faults in each drive.
3. If necessary, we can recommend some data recovery tools (they may NOT be free) you can download and buy that would allow you to copy files from the old HDD units to empty space on your good drive in the functioning machine. Of course, that will mean that the working machine's drive will have to have empty space to accommodate those files.
Let us know how this goes, and we'll help you through the next steps