Hey there,

A quick way to increase your iLvl would be to buy a crafted item from the market board (Or get a friend to make one if you know any crafters), this would allow you to continue the story chain if you're locked behind a dungeon and can go no further. Alternatively running the dungeons you have access to upgrade as you suggested yourself. Also tomestones. If you're collecting Mendacity by running dungeons you should have a nice little pile you can exchange in Ralghas for some nice gear that will last a while. Make sure you don't neglect doing tirals either.

Once you've gotten past at least 305 iLvl make sure you unlock the raid The Royal City of Rabanastre. This will allow you easy access to iLvl 330 gear.

If you're only just starting to do crafting I highly recommend leveling them up all at the same time as they have cross class skills that come in handy. But be warned crafting is a huge time sync and not for the faint heated.