I'm in the same boat with my old laptop. I downloaded Clonezilla
which will create an image of an entire drive and copy it to a
new one.

Haven't tested it yet, as I have to move some other hardware around
but it seems simple to use at first blush.

I've discovered that at least with Ubuntu it seems pretty tolerant
to being transplanted to different hardware. I had an old hard
drive with 09.10 on it from another desktop PC and it booted and
ran in an entirely different machine. I would still back it up
first but it wouldn't surprise me if you could install your old
hard drive in your new machine and have it boot, or at a minimum,
have your partitions mountable and data present.