1 - Prepare

Your /home partition isn't going to be reformatted if everything is done correctly, so if you know what you're doing you shouldn't even need to do a full backup. However, you've probably got some other things on the root partition that have been changed that you want to keep, at least for future reference.

# Backup /etc
sudo tar czvpf /home/etc.tar.gz /etc
# Backup root's home directory
sudo tar czvpf /home/root.tar.gz /root
# Backup anything else you might want to /home, for example MySQL databases
# (maybe backup /var the same way as above?)
# ...

Next you need to make a note of which partition is which so you don't accidentally format the wrong one.

$ df
/dev/sda5             11718932   2419420   9299512  21% /
...
/dev/sda7             44133352    219500  41671984   1% /home

So in this case, /dev/sda5 is what you'll be reformatting and installing your new operating system on, and /dev/sda7 is the one you don't want to touch in any way whatsoever.

Next you'll want to write the fstab line for your home partition somewhere so you can use it on your new installation. The following command will get that from /etc/fstab for you:

$ grep "/home" /etc/fstab
/dev/sda7      /home      ext3     defaults,relatime      0     2

The final preparation step is to move your old home directory somewhere. To do this you'll need to be logged in directly as root. If you're running an Ubuntu-based distribution and haven't set the root password before, do so:

sudo passwd

Log out of your graphical session, switch to a virtual terminal (e.g. press Ctrl-Alt-F2), and log in as root. Move your old home directory:

mv /home/alan /home/alan-old

Everything should be ready now - time to reboot into the installer of your distro of choice!

2 - Installation

Just install as usual, feel free to obliterate your old root partition but make sure you don't touch your home partition! Install as if your home directory will be on your root partition along with everything else. (It's possible to just use your old home partition at this point, but most installers will format any partition you choose to use by default. This way, there is no risk of that happening.) Reboot into your new fresh Linux installation.

3 - Juggling

Now all that's left to do is get your home directory onto your home partition, get your home partition mounted and move all your files back to their rightful places.

Yet again, this process will require you to log in directly as root to move the home directory around, so set the root password as before if necessary for your distribution.

Now log in as root on a virtual terminal (e.g. Ctrl-Alt-F2) to mount your home partition somewhere, copy your new home directory across, and unmount it again:

mount /dev/sda7 /mnt
cp -a /home/alan /mnt/
umount /mnt

Now we want to make sure the home partition get's mounted at /home as it should. Edit /etc/fstab with your editor of choice and add the fstab line from earlier. To make sure everything is correct, mount the home partition in a way that'll use fstab, and check that both the old and new home directories are there:

$ mount /home
$ ls /home/
alan   alan-old

Just in case your user ID has changed, make sure your old home directory is owned by your new user:

chown -R alan:alan /home/alan-old

At this point, you should be finished with using the root account, and can log in as normal. If you want to lock the root account (in the way Ubuntu does by default), run:

sudo usermod -L root

4 - Get your files

Now you should be able to browse both your new and old home directories using your file manager of choice. Open one for each, show hidden files, and start dragging stuff around!

This step might take some time, but has the added advantage that you get to learn where every application stores its user files, and also gives you the opportunity to clear out some junk you didn't need, clean up your directory structure, or whatever else.

Final notes

This process is probably slower than an upgrade, especially considering the step of sorting your old files into your new home directory. However, it gives you very direct control over the process, lets you start with a nice clean copy of your operating system, and if done correctly doesn't require a full backup to be made.

I much prefer this method to doing an upgrade because it leaves you free of the "old version cruft" that accumulates when you upgrade---an operating system upgrade is a complex process and is never going to handle every edge case.

If you trust all of your applications to handle old configuration files etc. correctly and you want to save some time, you can skip all of the steps relating to moving/copying of your user's home directory, so that when you mount your home partition your old home directory is used. If reinstalling rather than upgrading, I tend to do this.