Never a better occasion for getting Windows installation experience!
It's advisable to disconnect your old drive before the installation. This way you won't have to choose between devices during the installation, so less chance for a mistake. And it won't make a mess with bootloading (if you have two drives, sometimes the bootloader gets installed on one drive and the system on another, and when the one with the bootloader gets removed, the other won't start).
Then onnect the new one directly,and go forward with the installation. What's the worse that can happen? You can start all over again.
A clean install is especially recommended if the "damage" in the old drive (bad sectors??) affected the system files.
After the successful installation, you can install the necessary drivers and software you use, and then you can reconnect the old drive, and copy over your own files with the standard file copy mechanism of the operating system.
For your own files (e.g. files that you created like MS Word files, or music files/movies etc.) it's quite simple.
If you have saved games, or browser bookmarks, you'll have to check the actual software's recommended method for bringing those over. For example browsers can export and import their bookmarks. So you are supposed to export them now, on the working system, so that you'll be able to import them to the newly installed one.
Transition between systems can be a little painful sometimes, especially if you are using the old one for some times, and you have everything in its used space. But in this case, you can go back (boot into) the old system later, in case you had forgotten something.