What the answer depends on is the OS you want to transfer (and the suggestion to upgrade to a newer, larger and faster Hdd is worth considering, so I'll come back to that). The Microsoft NT-based OS's have a "Repair Install" that is more or less ideal for your purposes. In all cases, if the new system has a similar chip set to the old MB's, there will be fewer glitches -- as in two different NF4 based systems, for instance.
With a Win9X OS, it was necessary to prepare the OS before ending the last run with the old system. By this, I mean you use the Device Manager to remove all of the hardware items that you aren't transferring or don't know will have an exact duplicate in the new system. There are many devices in there that are built into your MB's chip set.
Next, if there was matching software for any of the devices, go into Add/Remove software and uninstall the programs. If you didn't follow those two steps with Windows98, you had driver conflicts and might never get the OS running right again. I never tried it with Win95 or WinME, however.
(Never tried it with Linux, either.)
Getting back to old hard drives vs. new: if you purchase a retail package for a new Hdd, you get a CD that will include software for copying the old drive's contents straight across onto the new one. If you prepared the old drive in the old system correctly, the copy would work the same way, and after the POST, the New Hardware Wizard would start on finding stuff. When it finished its first round, and finally loaded the desktop, your first action is to first run the CD for the new MB's driver install, and then run the video card CD.
With either of the current NT-based OS', the Repair option would handle much more of the process automatically. You still need to run the installs on the MB CD and video CD as soon as the desktop finally loads.
