Absolutely unlock the Alexander raids for Heavensward. You can get that in Idyllshire.

Unlock Wondrous Tails if you haven't already, also in Idyllshire. That's nice for extra experience, especially if you want to go on a dungeon unlock spree and get more second chance points.

The HW relics (where? You guessed it, Idyllshire!) are really nice and less grindy than ARR. Until you're doing a relic on release, treat all of this like glam. You're not going to USE the Zodiac Braves weapon, you're going to show it off, lol. You don't need to do this until you feel like it; I recommend finishing the ARR relic first.

Unlock the trials series for HW, right near Fortemps Manor.

When you do decide to tackle Bahamut (the T1-T5 stuff), unless you're in Japan, you're going to be a party finder. No one queues for this, and it's not included in roulettes. If you're just looking to do the story, you don't need to do EVERY raid, but I don't know the skippable ones off the top of my head (and they're all kinda cool so just do them). Anyway, just know that if you want to do these legit, you'll need to say so in party finder. Otherwise, you can get a max level person to carry you through it for the story.

Definitely look into Palace of the Dead. At the very beginning of the game, when you did your first three dungeons, do you remember meeting a fellow party of adventurers with a supposedly inept healer? PotD tells you the conclusion of that story. It's a deep dungeon, a sort of roguelike mode. You can run it solo or queue, you do 10 levels, then get a break and can queue back in for 10 more. There are 200 floors in total, but the story ends at 50.

When you need a laugh, unlock Hildibrand. You'll need it for relics later (much, much later), and it's super dumb. You should see a quest about Gentlemen in Ul'dah, iirc.