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.

I've used an Ubuntu based operating system on a portable USB external drive
to boot up with multiple computers. It worked fine but took a very long time
to boot up, probably because it had to redetect the hardware. So I imagine
your clone would be usable on other machines. After the initial detection
the boot up speed should be normal.

You can clone the drive, but I can think of good reasons not to go this
route. If your computer is the standard setup with Linux on one partition,
now would be a good time to make three partitions, a small one for swap (not
to exceed twice your RAM), a larger one for / or root (which is where Ubuntu
goes, 8 -20 GBs depending on space available and your habits) and the
largest one fore /home (where your user data goes). If you already have this
setup then try cloning, but I find there is nothing like starting with a
clean slate. Any problems that you encounter on new hardware are easier to
diagnose. The other way you will wonder if you imported problems from
settings meant to work on different hardware.

A fresh installation need not be hard if you: 1) save your sources list if
you have outside repositories enabled 2) create a list of packages installed
and 3) back up your home folder to a usb dive or key. Then
post-installation, you just add the old sources to your new sources list,
install new packages based on the backup list (automatic with Synaptic or
apt-get; it just reads the text file and downloads new ones) and copy your
contents of the old home folder into the new one. If you have a slow
internet connection then you can use aptoncd to save your existing packages
to a CD or DVD then re-install from there.

I think I will try the second option of backing up the old
home folder.

Sorry if this is too late to be useful for you. There is a program called
Remastersys which can create an ISO of your current programs. You can install
the ISO on a different machine, and have the same programs immediately
available, and all up to date.

However, the ISO doesn't include your data -- which includes configuration
files. Assuming the data is too large to fit on removable storage, you could
Share your home folder and copy things over across the network.

Right now I am fighting over why K3B is haven a fit on burning the ISO.
That is OK. I got a USB 8gb and going to go through that route. The rest I am
going to link up the old system to the new and start transferring stuff. The
newer system has more puff and go to run the new Gnome. My old Celeron is 7
years old and time to recycle it. After I wipe the system clean.  So how I
feel compiled to dive in the shallow end of the pool. Just because the warning
sign tells me not to.

If you are thinking of throwing away the old computer you can donate it
here. All you need to do is install a Linux distro and the recipient pays
for shipping.

http://freelinuxbox.org/

Oh I was not going to through it away. There is a company that recycles and is
a short drive away.
http://www.mcbia.com/
They will take old PC's and cycle them back through or part out. Safely recycle
the system to prevent going through the land fill. They do charge to deposed old
CRT and tear them apart where every thing is recycled.
I got the new system up and running and man. Does that make a world of
difference.