I am hoping to get the Deathwing mount before the event times out. However, my item level is ~280 and I am less than half an experience bar from level 120. What is the standard approach to getting better gear?
This thread should help. After you go to Nazjatar continue on to Mechagon as soon as possible. The main story quest gets you a 405 trinket with three sockets you can add punch cards for stat boosts. See if you can get in some mythic dungeon groups in premade group finder. Mythic +0 drops 400 gear.
At 120, farm Benthic gear in Naz’jatar. It’s 385 base ilevel and has some interesting effects. You can pick up Benthic gear for all slots except rings and trinkets. This will be a good method for you since you’re just going for ilevel, not trying to farm the best traits. You’ll get one free piece of Benthic gear just for doing the unlocking quests for Naz’jatar.
For the ring and trinket slots, Heroic dungeons should give you a chance at some decent ones, or world bosses, world quests, and emissaries, although those are unreliably generated.
Basically if you just keep questing and doing some dungeons you’ll have it in no time. At this point catch up mechanics make it really easy.