Jayce is a Mid and End game character, that's why during early game you have to play pretty safe.
Indeed you'll do low damages until you have at least your sheen and your bloodthirster fed, that's why you've to play as a distance whenever you can. Anyway, you'll not deal high damages, you need help at the beggining to kill or an opponent who's taking a lot of risky actions.
If you're taken into a melee fight, switch your weapon immediatly. Use Q then E to put your opponent out of the fight. If you choose to chase him launch W and follow your opponent in melee fight then use Q and E when they're up.

As a conclusion, you have to play as an AD range during early game, focusing on minion farming. A little further, if your opponent is going back during a 1v1 fight and if he's low, you can switch to range and use E then Q on him to pick up a safeful kill and go back to you turret.

After you've end your trinity, you'll deal really high damages, you've to focus the ennemy carries to carry your team to victory.
Firstly, you have NOT to engage fight, you're not a TANK at all ! Stay near of your team and try to follow her by coming by the back or the side of ennemy team as a range damagers. Here, you'll have to launch E then Q on ennemy team to do some damages and take a movement boost. Then use W before switching weapon and jump with Q on the biggest carry of the ennemy team. Then use E and W and continue to harass ennemy carries if they aren't dead.

Don't forget that when you've your hammer you gain armor et magic resistance bonus that's why during a fight going back to your crossbow can make you die pretty fast.

Hope you'll be good enough to handle Jayce and see you soon, on EU West server !