The truth is that what you "should" be doing is what interests you.

The only thing that everyone should do is the Main Scenario Quest (the fireball icon quest). This should appear at the top-left under "Scenario Guide". This unlocks new areas and its associated content.

After that, you can do things unlocked by Blue Sidequests: High-End (raid) duties including their 4, 8 and 24-48 person varieties, Field Operations (large exploratory content), PvP, Hunts, Treasure Maps, sidequest dungeons, Deep Dungeons, Variant dungeons, level other jobs that have different playstyles, crafting/gathering/fishing, play games at the Gold Saucer and gather Triple Triad cards, rank up Beast Tribes and Custom Deliveries, complete all the Role Quests, maybe complete all side quests if you are into lore, do social activities such as roleplaying and going to venues at player houses.

If any of those things specifically interest you, you can of course ask here for specific information about it when you're ready to do it. There are different types of players, and often they are only interested in certain things. Do you want solo content? Do you want social content? Do you want challenging content? Working out where your general interest lies will help us make suggestions.

Square Enix has a manual at: https://na.finalfantasyxiv.com/lodestone/playguide/

You can also type /achievements to see what you haven't done.

You can type /journal then click Complete to see what quests you have completed (it also explains what happened in those quests). You can replay a lot of the past story by going to Western Thanalan and doing the quest that unlocks New Game+ to replay it in chapters. Or you can just re-watch cutscenes at "The Unending Journay" at an Inn. You can find an Inn if you /teleport to Gridania and go to the airship area.

If you get stuck on a quest, you can usually look up the quest name followed by "ffxiv MSQ" on YouTube to find someone who played through it, and you'll see how they got through it, although normally it's intuitive enough.