You really need to state what the new pc supports. If you are a novice at pc repair we can rule out fixing the boot loader so the HDD will work in the old pc. Better safe than sorry, you don't want to worry about losing the data.
You need to open the case and see what the new pc uses. Hopefully it will support both SATA and IDE.
If the new system HDD is IDE as stated you can fit the "data" drive as slave. You will most likely need to change the jumper (lil plastic bit) at the back though. Note what the "new" drive is set to. Would be best to set both to CS (cable select) then plug the "data" drive on the available connector below the "new" drive.
If the "new" system is using the DVD drive on the IDE this is ok as you can swap the cables to the HDD. You'll lose the DVD obviously till the cable is put back.
A caddy is an option, if you go this route ensure that you buy one that is IDE, most will be sata i think you'll find.
I've made this sound more complicated than it really is, take your time and think. Ask if you have doubts......there are no stupid questions, just stupid people who dont ask questions ;p