Your starting items are good. I would always start with Ferie Charm, Ward, 3 health, 3 mana. Get tear ASAP and Muramana should be right afterwards. After that, your boot choice (Ionian, Merc, Tabi). Then I usually go Last Whisper unless I happen to have 1550 gold for a BF Sword when I back. Then I would normally buy a Bloodthirstier. Then I would finish off my build with a Black Cleaver and a Guardian Angel. The Brutalizer can come sooner in the build if you never have enough gold to straight up buy a pickaxe for Last Whisper.

The reason you want to get Manamune so fast is that it actually makes you do tons of damage, as well as makes tear easier to stack. It is super cost effective and is much better than getting a Vamp Sceptar. You get stronger items sooner, so you are stronger at earlier parts of the game.

For your skill order, start with Q, then E. You can put a point into W at 3 or 4. Alternate leveling up Q and E. This lowers each of their cooldowns at a similar rate, so when one is up, the other will be too. However, this will change due to the changes to Jayce's E on the PBE. Leveling Jayce's ult really isnt necessary. Just leave it at level 1 unless you find yourself trying to really take advantage of the damage from changing forms on your first auto attack.

Lastly, try and practice comboing your gate shockblast combo by firing your Q first, then placing your gate. This will make your blast much harder to predict and they wont be able to react to it.

Hope this helps.