Welcome to Jayce and be prepared to have a sh!t ton of fun. Jayce is part ranged and part melee. He initiates with his ranged form and then finishes off the enemy as melee. Because he has increased defenses in melee form, when you're being hit and you can't kite, go melee to soak up some of the damage.

When you initiate you want to start with your ranged Q E combo. This'll do a lot of burst damage and proc furor. If you can keep hitting your target with ranged, use W and go for it. If they're getting away, pop ranged W then switch to melee. If you're close enough to get a hit on, E then Q to catch up. Melee W should be activated during the entire melee fight.