There's multiple major things to do after MSQ is completed, but in terms of combat the main ones are
1.Normal/Savage Raids. Specifically the Eden raid series for this expansion. 3 tiers of 4 bosses spread out over the course of the expansion. The final tier will drop with the next major patch. Savage raiding is considered the "true" endgame for hardcore players, it's the most difficult content in the game (aside from Ultimate tier, but that's basically bonus). Normal difficulty is where the story is at though, so anyone should be able to experience that. Savage is largely for the challenge, Best in slot gear, and dyable versions of the normal tier armor.

2.Alliance raids, specifically the Nier raids this expansion. 28 person raids, pretty self explanatory. When current, the gear they drop serves as catch up gear (not quite as strong as current Best in Slot gear, but only about 5 ilvls off) and you also get coins which can be used to augment the tomestone gear (which otherwise can only be upgraded with savage).

3. Trials. Each major patch aside from .1 has a new trial. The .3 patch's trial is connected to MSQ, but the other trials are part of an optional storyline, with this expansion being the Weapon robots from FF7 (so far Ruby weapon and sorta Sapphire weapon in a solo fight). Normal difficulty is part of the story, extreme difficulty is 100% optional. Extreme is a middleground between normal difficulty fights and Savage. It requires fight knowledge, teamwork, and proper understanding of your class for a certain level of DPS, but it is not nearly as deadly or intensive.

4. The relic weapon. Every expansion has a relic weapon that you work on over the course of the expansion. This expansion is a bit different in that there's a prerequisite to even start the series. You need to do the Stormblood alliance raids (Return to Ivalice) first to unlock the relic weapon. Relic Weapons are generally largely about the glamour, they become Best in Slot at the end of the expansion, but before then they tend to fall into the "catch up gear" territory. Relic weapons tend to be something of a grind to get, but they rarely require much active skill, just patience.

Aside from that, there's crafting and gathering. Each crafter and gatherer class acts like combat classes in that they start at level 1 and have to be leveled to 80 and have their own tools and gear with stats that you upgrade as you progress.