Keep in mind that "setting it to slave" isn't the only thing you need to do when hooking up your old hard drive to the new computer.

Check your existing drive - if it's set to "Master", then set the old drive to "Slave".  The master drive should be on the end plug and the slave on the middle plug.  If your computer was manufactured sometime in the past decade, just set it to "Cable Select" and don't worry about it.  The drives will set themselves to master or slave based on which plug they're connected to.

If you can afford it, get a computer that supports RAID functionality on the motherboard.  Then pick up a second hard drive that's the same size or larger than your existing one and use the RAID functonality prior to start up (or software that comes installed) to create a mirror.  That way, if either drive fails you still have all of your data and giving you time to replace the bad drive.  Once the bad drive is replaced with a good drive, all your data is automatically copied over onto it to maintain redundancy.

Myself, I have to have backups.  There's something about me that drastically cuts the life-span of a hard drive located within a few feet of me.  I have a stack of drives next to my desk six high that failed less than one year from the manufacture date.  I currently have four 1 TB drives operating in a striped-mirror arrangement with backups being performed nightly onto another computer.  This backup computer, understandably, is located in a different room I don't frequent.

@ The good doctor:

After this story your username is starting to make sense to me...