For Darius Trifarian, can he trigger his effect if he's the second card that gets played or does he have to be played first?

Yes, Darius, Trifarian can trigger his ability when he is played as the second card in a turn. He does not need to be played first.

According to the RiftJudge FAQs regarding Darius, Trifarian:

• Darius counts himself as a played card. If he is your second card played this turn, his ability will trigger and he will ready himself.
• Sequence:
  1. Play your first card.
  2. Play Darius as your second card.
  3. Darius resolves and enters the board.
  4. Because he is already on the board when the "after you play a second card" ability checks, he sees himself as the second card played.
  5. His ability triggers, giving him +2 Might and readying him.

Important Nuances:

• Darius must be the second card: If he is played as the third, fourth, or later card, his ability will not trigger because the "second card" condition has already passed.
• Tokens do not count: Playing a token does not count toward the "second card" condition. Darius must be the second actual card played in the turn.
• Already on board: If Darius is already on the board from a previous turn, he will also trigger his ability when you play your second card of the current turn.

Source: RiftJudge FAQ #7647, #1119, #4088, and #8290.