Let's see, your options are:
Exping: the act of gaining experience, be it through quests, missions, puzzles, treasures, or just by killing opponents. EDIT: disregard the previous term, as it looks like it's not as widespread as I thought. Consider it just a localized version of leveling.
Leveling: the act of gaining levels, usually by accumulating experience points (XP). Generally used refering to the actual moment of going up a level ("He just leveled up"), can also be used for a continuous action ("I'm spending my time leveling")
The term "Farming" instead refers to the act of going through the same actions for long stretches of time to hoard single resources (you can farm money, minerals, items, and even XPs)