Level 16 and no quests?

[Felis]

I am level 16 in the Limsa Lominsa starting zone and I haven't had any quests for 2 levels now. My guildmaster doesn't have my next quest available until level 20 and I can't use the airship.

I would say you miss somewhere the next mainstory quest. You should get with level 14 or 15 a mainstory quest which give you the airship pass.

Look in the archiv of your quest journal what your last story quest was and then visit the last mentioned npc.