The main scenario quests have a different sort of quest marker. The side quests (not part of the MSQ) have a marker that looks like a Q with an exclamation point inside. The MSQ quests have a round marker with what looks like flames coming out of the upper right side, with an exclamation point inside the circle.

Concentrate on the main scenario quests, and you will unlock the dungeons and the chocobo mount, though not necessarily at the intended level.

For example, my husband did so many side quests and fates and whatnot while leveling that I think his character was level 23 before he got through the main scenario quest to get the chocobo, and he was above the level for the three dungeons and the primal quest when he did them (though that's not an issue because your level is automatically synced to the level of the instance if necessary). If that sort of thing happens to you, you'd still be able to unlock the dungeons and get the mount even if you're on the free trial, you just wouldn't be leveling while doing the quests since xp is locked at level 20.