I know you're not allowed to play them on the 1st turn but does this mean the 1st turn of the player who starts the game or both 1st turn for both players?
Yes, the player going second can play Trainers/Supporters/Stadiums. However, both players can't evolve on their first turn (unless the second person has Rare Candy/BTS/evolve from attacks/etc.)
And also, when using cards like Porygon 2, and sayble, you are aloud to use their attack/power since it is not the normal means of using a trainer card.