Both of Jayce's Q is AoE, so if you Q a minion an enemy champion is standing next to, it'll hit them for the full damage & slow!
Jayce's Shockblast can be buffered, which means you can cast other abilities during its animation, which means rather than placing your acceleration gate first, and then shooting your shockblast through it, Q and then E directly in front of you for an incredibly fast empowered shockblast! Also, much like Urgot's E - Flash, you can Q - Flash for an almost impossible to react to projectile, and if you're fast enough, empower it with your gate!
Hypercharge is an auto-attack reset, so you can AA - W - AA - AA - AA for an incredibly fast 4 auto combo! Also, your W does not end if you switch forms, so you can get 3 very fast melee autos, which is useful if you're running Conqueror, or taking early tower plates!
In melee form, if you input an auto-attack command while in range and then quickly knock the target back, your E animation will go through first, and your auto will follow them well outside your actual auto attack range.
Buffering Q and placing the gate second can feel very janky at first, but if you place your cursor over where you want to shoot, start the shockblast, quickly sling your mouse back in front of you, and then place your gate, you will become MUCH more accurate.
Probably one of the best trading tools is using an accelerated shockblast on a minion or group of minions your opponent is right next to, evaporating the wave AND like a quarter of their health bar. Watch them seethe and mald and cope and sneed, you're Jayceing now!
Both Shockblast & Acceleration Gate grant vision where they land. If you shockblast a jungle monster, or place an Acceleration Gate over it, you can switch to melee form and leap onto it, effectively creating a wall hop! Try it when you're running away in a lane, or chasing someone into the jungle!
Jayce's E is now a proper knockback, and while that removes a lot of anti-dash potential he once had, he can now E-Flash to properly insec! Do with that what you will.
Your melee Q animation can be cancelled by any emote (Laugh, taunt, dance, etc). Not in the sense that it saves any time, rather think of Zac's jump sometimes being a very fast walk. Using this, Jayce's Q animation can seem much less obvious to the enemy, and can possibly give you the edge.
Jayce's Acceleration gate gives you a large, but decaying movespeed buff when you're on top of it. If you click just slightly to the SIDE of Jayce, you can place the gate vertically and run alongside it, refreshing the max movement speed for a few ticks. This may seem like a small thing but it really matters when avoiding ganks, that extra distance you gain could be the difference between dying and not.
Jayce has a combo specific to the rune Hextech Flashtraption. If you melee Q while channeling Hexflash, the channel is canceled. Unlike the animation cancel with the laugh emote, the channel cancel allows you to use other abilities in the middle of your Q cast, meaning you can start your E cast while you Q. Chain this with the AA input "ranged melee" mechanic, and that's your entire melee rotation. The kicker is, if done correct, this combo effectively outputs your entire melee rotation in 0.04s; Even though at lvl 4 with just a longsword it does around 300 dmg, according to a training mode dummy, the combo will read as somewhere around 6300 DPS.
Do keep in mind though, Hexflash can only be used out of combat, so it's best to be used out of a brush.
The full combo, Hexflash-Q E AA, is called a Hextap, and there's basically no counterplay to it.
(I included this here rather than the "Combos combos combos" section because of the unique inclusion of Hexflash, and how deep into explaination I went.)
Jayce can shoot 2 shockblasts through 1 gate with enough Ability Haste. There are 2 major breakpoints for this, one at 95AH which requires some finnegling to pull off, and one at 120AH, where Jayce can comfortably place his gate point blank for 2 shockblasts. At 95-119, you're gonna have to shoot the shockblast, wait for it to reach about the max length you can place your gate, and then place the gate to empower it, walk up to the gate and wait for your Q CD to come up again, and then shoot your 2nd one; Placing your gate as late as humanly possible is what allows it to be still up by the time your next Q comes up, but at 120AH that just doesn't matter.