Honestly, there is almost zero serious side content before you hit level 50. There are things you can unlock such as using your chocobo in combat and glamour and stuff like that along the way to 50, but there isn't any important story content on the side to worry about until you hit level 50. After you reach lvl 50 and complete the second 8 person dungeon, some side things unlock.

One of them is mandatory to finish before you can move onto Heavensward, the Crystal Tower alliance raids. Don't let the word "raid" make you nervous, it's almost 100% faceroll difficulty at this point due to general overgearing, but it's still pretty fun. Another (totally optional) thing that unlocks that is a bit more complicated to complete is the Binding Coil of Bahamut raids. These unfortunately were the game's hardcore raids, and as such you won't find any parties for it in duty finder. You can get a group together to breeze through it if you put up a group in Party Finder and ask for help to get through so you can see the storyline, with a couple of lvl 80s and you can blow through it easily. But it's also completely optional if you don't want to bother with it, and it isn't going anywhere in case you may wanna come back to it later. Thirdly, there's a comedy questline about a detective named Hildebrand that unlocks in Ul'Dah. Fair warning, the questline is a total slapstick farce. A lot of people really like it, but at the same time some don't because how tonally different it is. But it also has a continuation in both Heavensward and Stormblood.