some questions

[OlyverW]

Each class has a hunting log you can find it in the menu. There are 5 ranks to it you get a new one for every 10 levels you complete. How it works is each rank has a list of enemies to kill after killing certian groupings you get an xp bonus and when you completely kill all the marks in that ranks log you get another xp bonus

Since you are a gladiator you can use this link to find your specific marks

ffxiv.consolegameswiki.com/wiki/Gladiator_Hunting_Log

[ExiledOne]

also as you travel around make sure to participate in any FATEs that you encounter, even if they're lower level, but take care in the higher level ones.
In fact, if the FATE is too high I would advice that you avoid it unless there's a large group also doing the FATE.