Yes transfer your fish from the old to the new as soon as you move the filter over to new tank. Remeber to run the new tank for a day or so to make sure everything is working right and temp is stable before doing the transfer. Id also tranfer as many of the decoration as you can over too. This should prevent a mini-cycle from happening in the new tank after the transfer.
side note: You could leave both filter on the tank as long as you want too. That way you always have that filter for the old tank ready to go if you ever need it for the old tank again.

Never know you might have an itch to have more than one tank going.