Yes you need to follow the main story and eventually it will lead you to the next expac. The only way you can get around not doing the story is purchasing the story skip items on the Mog Station for real money, but I'd advise to just stick at it for now. It'll get better.

Job abilities past level 50-60 are locked behind quests so until you get access to Heavensward content, unfortunately you can't use any further job actions between those levels. Stormblood is an exception though as you can learn job actions just from leveling up (so 61 I think onwards), but the level 70 action will be locked behind the job quest.