You should be doing any quest with a + on it to unlock new things. There are a lot of raids, trials, and optional dungeons to unlock along the way for more story and which can be fun to do. Also, almost everything is still doable in duty finder, so no worries about it being old content. The exceptions are: The Binding Coil of Bahamut (The first 8-man raid), the Savage difficulties for the raids in Heavensward and Stormblood, and Extreme difficulty fights (Marked as such or with "The Minstrel's Ballad".) That said, these are all perfectly doable in Party Finder, though you may have difficulty finding folks for the HW and Stormblood savage raids (But there's also less reason to do them aside from seeing new mechanics to fights you've already done)

Now, that said, if you're just interested in getting up to endgame levels ASAP, it's basically just MSQ and job quests to 80. But all the stuff along the way is the same stuff you'll be doing then, so don't feel like you have to rush