When playing a Duration Card in Dominion-Seaside, is that Duration Card your 1st action of your next turn or can you play a card from your hand 1st and then play the Duration Card?
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).
The effects of duration cards happen first. Only after resolving these effects can you begin to take actions.
No, it is not the first Action of your next turn. You are not playing the Duration card a second time. You played it last turn. From the rules:
Duration cards are orange and have effects that continue past the end of the turn in which they were played.
Additionally, if you check out the Duration play example in the rulebook (pages 5-6), it shows getting the effect from a Duration card you played last turn, followed by playing another card as the first Action.
All effects that say "At the start of your [next] turn" trigger before anything else you do. This includes Duration cards, Horse Traders (if revealed in reaction to an Attack) and Prince. However, if you have multiple of these effects triggering, you choose what order they take effect. Duration cards do not count as an action played that turn (which is important for Conspirator), but they do count as an Action card in play (which is important for Peddler).