Move the old filter along with the fish to make sure the tank doesn't cycle all over again. If you are getting a larger filter, run it alongside the old one for several weeks before removing the old one. To reduce stress you can also add some of your old tank water. Make sure the temperature is right in the new tank, acclimate the fish in a bucket with their old water, adding new water a little at a time until it is close to the same as the new tank, then net them in and you're done.
EDIT: This might sound obvious, but make sure you use chlorine/chloramine removers on your new water before adding the old filter.