You can check your EXP progress at the bottom of your screen. There are red, blue, and or sometimes yellow arrows above the bar, telling you that you can upgrade your Player Stats when the golden color bar reaches it through.
When you first start out, it is very quick to level up. Finishing up quests give you a lot of EXP continuously. At some point, usually around level 55-60, the curve slows down quite fast to the point where quests isn't enough to level up continuously. This is where dungeons will be a big part of gaining experience points.
After around level 83, it gets even more difficult to level up, taking a lot of effort to reach level 85, when the last maps (Chronology Forest and Tempest Desert) are unlocked. At this stage, the player will need to do dungeon runs constantly to level up.
Professions as Archaeology, Fishing and Guardian Knight has a separate EXP curve to the normal experience points. These kind of experience can be gained by performing their corresponding activity, like fishing a lot or gathering a lot for instance.
When reaching specific levels, you will also gain bonus stats or extra traits.