Generally speaking, there are three main builds for Jayce now
1. The BT/BC Rush build. This build you pretty much rush BT and BC. You go something like boots->brutalizer->BT->BC or something along those lines. This build gives you the most late game damage the fastest, but is a relatively weaker compared to the Triforce build during the early and mid game. Essentially, you're building Jayce as an ADC. Relatively weak early game, strong late game.
2. The Triforce build. This build is the oldest one and the one I still favor. You give up a lot of raw damage compared to the BT build, but you make up for it in added utility and early and mid game power. Triforce, despite its nerfs, is still a superior item for the early and mid game compared to BT. You get health, mana, ad, attack speed, crit, slow, sheen proc, and MS. All of that is highly useful to Jayce. Typically with this build you go from boots->2 dorans blade->phage->triforce. After Triforce, people typically go Bloodthirster or other damage or defensive items. This build has the strongest early and mid game.
3. The manamune build. Imo this is the worst build out of the three. It gives you a LOT of mana so you can basically spam your poke forever, but it gives you the least damage output, the least survivability, and the stats it does give is highly inefficient on Jayce. It does not give you the damage of the BT build, nor the utility of the Triforce build. Also, Jayce really doesn't need that much mana. I myself hardly ever have a problem with mana with just the sheen from the triforce build.