The software you want is only good if you have your entire iTunes library on your iPhone (songs, movies, ringtones, apps, ebooks, photos, podcasts, everything). Otherwise you won't get everything transferred anyway.
The best way to do it is get hold of a removeable storage device (like an external HDD, USB stick, burn them to a DVD, or remove the HDD from the old computer and plug it in to the new one alongside the existing HDD) and transfer your iTunes media folders using that. When they're on the new computer you can drag and drop the folders into iTunes.
What you do need to do is de-authorise your old computer first, then authorise your new one, otherwise syncing will prove very difficult as iTunes will tell you that your iPhone is synced with another computer.
http://support.apple.com/kb/ht1420