If I play Darius trimaran as the second card of my turn does he enter readied and buffed?

Yes — playing Darius, Trifarian as your second card of the turn both gives him +2 Might this turn and readies him.

His ability is "When you play your second card in a turn, give me +2 might this turn and ready me." A card counts as played only once it has fully resolved and entered the board, so Darius is already in play when the trigger checks — and he sees himself as that second card. (FAQ #1119, #6776, #2483)

One clarification: that's +2 Might, not a Buff counter. No Buff is placed, so effects that care about units being buffed won't see anything here. And per CR 416.1.c, if he's somehow already ready when the ability resolves, the ready portion simply does nothing — the +2 Might still applies (FAQ #1904).

Also note: if he's the third card of the turn, the ability does nothing, because he was never on the board to "see" the second card.