For any quests it's enough to not stand in telegraphs and kill things in a certain order, such as killing casters first.

For fates it can vary. If it's just a fate with loads of things to kill then not pulling more than two at a time usually works fine.

For boss fates it's very common to be unable to solo them as they are designed for multiple people. If the boss is melee it's usually easier because it can be kited.

Believe it or not tanks and healers can have a hard time soloing too. While they have more defensive abilities to draw from, they lack the damage an actual dps has. They often die simply because the boss's hp is so large that the fight lasts too long and they exhaust all their defensive options too early in the fight.

Players often use chocobos to make solo play easier. They're not very good tanks unless given very specific food (which can only bed fed at a stable) and they will need healing like any other tank. However they're pretty decent healers and dps, and can make a very significant difference in how much content you can solo.

The best way to take advantage of chocobos is to start early. Although the chocobo syncs with the level of content you're doing, a rank 1 chocobo in lvl 30 content is much more effective than a rank 1 in 50+.