If you wanted to start from scratch, then yea, a new computer would help you "move on". But it wouldn't help with the backing up of stuff.
In Safari, there's a way to export your bookmarks. You should be able to backup all your music, pictures and documents easily as they should all be in your home folder. Finally, you may want to make a list of programs you have installed that you will want to download and install later.
Since your hard drive is failing, you will essentially have to reinstall everything from scratch. Unless you decide to ghost the drive, in which case it's as simple as creating the hard drive image, storing it somewhere such as an external hard drive, and then "imprinting" it onto your new drive.