Will Jayce brilliant inventor trigger on itself and the first non-toke gear played each turn?

Yes — those are two separate trigger conditions on the same ability, and both can fire in one turn.

Jayce, Brilliant Inventor reads "When you play me or the first time you play a non-token gear each turn, you may ready something besides me that's exhausted." Playing Jayce fulfills the "when you play me" clause, and since Jayce is a Unit, not a Gear (Rule 148), that play does not count as — or use up — the "first non-token gear" clause. If you then play your first non-token gear that turn while Jayce is still in play, that clause triggers separately.

So in a clean sequence — play Jayce, then later play your first non-token gear — you may ready once per trigger, up to two different exhausted things (never Jayce himself). Each individual trigger still only readies one thing. Note the gear clause can only fire if the first non-token gear is played after Jayce is on the board; a gear played before he arrives doesn't set it off.