Duration cards in Dominion - Seaside - Are they your 1st action of your next turn?
Note that the cards say "at the start of your next turn". So you must obey the text literally; you do what it says "at the start", which means before you do anything else. If you played multiple Duration cards last turn, so that you have multiple "at the start of your next turn" things to do, you choose the order in which to do them, though there are not any Duration cards for which that order would matter. However, this could matter with the promo card Prince.

A couple other things to clarify:


You don't need to use up your 1 action per turn for the next-turn Duration benefit. You can still play your normal action after resolving the Duration.
The Duration card is NOT played on your next turn. It will not count towards the number of cards played that turn, which can matter for Conspirator (from Intrigue). It will, however, count towards the number of action cards in play, which can matter for Peddler (from Prosperity).