Hello,
I know the feeling when a HDD is about to die
About your question.
For _ME_ the easiest way is to determine what files i actually do want to keep.
From your question, we as readers do not get to know if you have more then one? Do you run your operating system on the same drive as you store all your other data/files?
IF so, then yes, the "easiest" way is to copy the data over to the new drive once your new OS is installed and everything is up and running. I would recommend have a separate hdd for your OS tough.
From own experience, a old drive usually tend to contain alot of junk files thats not needed when you install a new system, so if you do know what you want to keep (passwords, pictures, gamesaves, application configurations etc) i would just copy over them and then re-install everything onto the new disk.