Replace hard drive & reinstall OS

I have an older system, and the hard drive failed a couple of the Dell diagnostics tests, even though I am not having any problems.  I've never changed a hard drive before.

I have the Windows Vista Home Basic reinstall disk that came with the computer, along with the MS Office Home & Student disk and a couple other program disks that came with it.

Unplugging the old hard drive and changing it out is the easy part; I'm confident about that.

My fear is that when I try to reinstall the OS and other programs on the new hard drive, Microsoft (and the other companies) will "think" I am trying to pirate the software onto a different computer.

How do I make this as quick, smooth, and painless as possible?  I can't afford to spend a lot of time trying to straighten out a screw-up.

If you can, please help.  Thanks!