This is my first build, I only played once with Jayce but i understand that this type of champion is an op hybrid from ranged to melee to ranged.
I hope to update this build asap but for now i only want to esplain the skill sequence for a fast harassment and why i chose this items.

First of all, all the abilities of Jayce works really well together there is no end to the combo you can Do with him if you play him well, So to level 3 you must have all his abilities without an exception, for a better harassment your Q is your base for damage and your E work very well in every case (ranged or melee). The W is not so powerfull as the others 2 but is useful for destroing turrets (in ranged) and doing passive damage (in melee) remeber that this effect continue also if you change your type of combat.
remember that every time you change your stance with R you gain passive bonus speed that can be usefull stacked with your ranged E for an easy escape or for chasing enemies

How to destroy your foes? nothing more easy. you start with your ranged stance when you want to kill someone start with closing to him not much far neither near (near the ranged minions is ok for a better aim and better chasing). Use your E in front of you, now use your Q your E is still active So use the bonus speed from this abilities and change asap stance (R) now Q on your enemy W and E in the opposite way of your enemy escaping rechange stance(R) use again you E then Q if he is still alive W (from ranged) and he will be death . If not do this one more time

Jayce is an hungry mana champion and at the beginning of the game he cant stay long in the lane if he try to use his abilities for killing the enemies and you can see that my skill sequence require a lot of mana use , so if you want to do the first blood or kill someone early game you must haveasap when you are at 607 gold minimum you must recall asap and start to stack the Tear with Blue bonus or Red this champion could destroy is foes incredibily.
why cooldown reduction? because whole the damage you do is for your abilities this is not a Tryndamere or and Ashe you must use your abilities.
I take Sheen because of is amazing effect that work well with all the skills and is usefull against turret. Then you must upgrade this into Trinity force because it gave all the passive that you need
after this 3 objects your foes must be scared of you, if not there is something wrong maybe they are building armor in that case use thismaybe because they resist more in lane than you.. ok it's difficult if you play this pg well but in every other case you needI must remember that this is an AD champion?
the other items is in base of your enemies , they are usually focus you? or you die because of low life?

this is my first guide so I dont now how to put masteries on this guide, but an offensive or utily chose of masteries could be well.

In team work there is nothing that you can you do of really important, only use the skill sequence that i have said and use your ranged E for helping teammates or your melee E
remember that your Q in every stance do massive AoE damage so use it!!

I usually dont use Runes, because of i still havent found a perfect pg for me. i think that attack damage runes or better cooldown reduction may work well

Ok, this is my first build I do my best for help you summoner ;)
please rate or leave a comment if there is something wrong or if you have to give me some gameplay clips or others .. thx all for reading this :)