STOP! If you connect old suspect hard discs to your nice new pc it might be that they have viruses etc. First thing before you start transferring old data is to run a virus check!
Now, how you do this depends on what the old hard discs are i.e. SATA or IDE. What size they are 3.5" (towers) or 2.5" (laptops). And then what you have in the way of a working computer now. IF the discs are older AND from a tower as opposed to a laptop then likely to be 3.5" IDE format, which look like this, generally a "big" white, 4-pin power connector and a "long" data plug often on a ribbon cable, sometimes a round cable:
Let's stop here for a second, if it, the disc(s) looks like the photo above then I have a 3.5" caddy that you simply put the IDE drive in. Connect the supplied mains psu to it and via a USB lead plug it into any pc of your choice. Dead simple and you won't have to open your existing pc. Downside it's a tad slow in terms of data transfer - more on that in a mo but TBH more than adequate for transferring a few files. Anyway, if you want I could post this to you to borrow no problems. Or dive down to Maplin and get one yourself:
USB 2.0 3.5 Inch Hard Drive Housing : 3.5 inch HDD Enclosures : Maplin Electronics
or
3.5 Inch IDE / SATA USB 2.0 Combo Enclosure : 2.5 inch HDD Enclosures : Maplin Electronics
It may be that your existing pc tower (assuming that's the case) has a motherboard that supports both SATA and IDE drives. You could fit the old drive in permanent / semi-permanent. This will mean opening the case, screwing the drive in and connecting an IDE cable and power supply to it. Of course it might be that you don't have IDE channels or if you do the leads aren't in there! Plus side is it'll be quicker on the data transfer than via the USB route. Again speed for what you want is not really an issue.
Then again even if your new pc only supports SATA then there are lots of gizmos that let you put an IDE drive inside a SATA only pc, like this:
SATA To IDE Convertor : Internal SATA Cables : Maplin Electronics
Of course you will probably need to do this for each disc so I reckon the caddy is the way to go!
Many ways to skin this cat!
BTW, SATA drives are generally physically the same size as IDE (3.5" or 2.5") but the connections on the back are different. Plus side you don't have to mess around with changing jumpers:
The above photos show 3.5" drives.
If you get stuck, shout. The nerds at Maplin would always point you in the right direction. Offer still stands on the caddy loan. Only using Maplin as an example plus they're generally local. (I use Ebuyer for most of my stuff BTW - cheap as but online).
Cheers