What lv60 HW content should I work on unlocking before I start up SB content at 61?

[Naria]

Unlike the Crystal Tower series Shadows of Mhach alliance raid series doesn't end up playing into the MSQ directly. It is worth doing and fills out several points of lore but at this moment isn't a hard pre-req to anything else. It does, however, have several optional pre-reqs itself. Namely it goes Aurum Vale -> Ampador Keep -> Lost City of Ampador -> Ampador Keep (Hard) -> Shadows of Mhach.

The Wanderer's Palace and Wanderer's Palace (Hard) also fill out some of the background for the raid but they aren't really the same story-line. The Lost City of Ampador (Hard) has a bit of foreshadowing for Shadowbringers.

I'd also do the Heavensward primal side quests series "The Warring Triad" (edit). Primals were into a separate questline instead of the MSQ starting in HW. The starting quest is "The Gods of Eld" in Moh Dhona. The intended order is The Gods of Eld -> The Diabolical Bismark -> Thok Around the Clock -> When the Bough Wakes ->etc. The first two fights are Extremes so you can skip straight to When the Bough Wakes and continue from there, but I'd recommend getting a lvl 80 friend or just putting a PF up and doing the quests in the intended order with an unsynced party. This quest-line compliments The Shadows of Mhach and has some foreshadowing for Shadowbringers.

There is an epilogue quest for the Dragonsong War that you can pick up at the Fortemps Manor.

There is another Hildibrand side quest series and The Saint Endalim's Scholasticate series but those are both fairly self-contained and require the completion of ARR's Hildibrand questline to start. Neither end up playing into the MSQ and are just for fun.

Hopes this helps!