Open Synaptic. Click on File, Save Markings. Check the box at the bottom to
save Full State. Give the file a name and put it in a place where you can
access it on another computer. Dropbox or a usb key works nicely.

To use it, Open Synaptic on the other computer. Click on Read Markings.
Locate the file. Then Apply the changes.

There are limitations. It only will do files in the standard repositories.
Outside sources will not work, unless you add the additional sources first.
If you switch distributions it will not provide all of the packages as
Ubuntu has more packages than many distros. Also it may install things that
you do not want or that are not available on another distro. The farther
that you deviate from the Ubuntu tree the more problems you are likely to
have. Even Mint can give you problems, but they are likely to be minor.
Debian is likely to be more problematic where even things like Firefox are
not in the repositories. Instead you will have Ice Weasel.