Down Time

The background, behind the scenes, or even just 'when in town' are often places referred to as Down Time. Down Time is a place where you can have your character accomplish special tasks for personal reasons / motivations. These tasks can be a lot of different things:
Find an Item: Want that special sword of lightning? Maybe find it off screen. Learn a Spell: Particular spell you want to learn just not showing up? Research it! Research an Area/Person: Heard about that place huh? Maybe go looking for it on your own.
These are not limitations, only ideas. Bring your ambitions to the table, or a message to the Dungeon Master and see what happens. Some of these things might take time, or some solo play sessions. What sets Dungeons and Dragons apart is that there is no ceiling on what 'could be'. So go find that magic staff, clear out that castle, start a village, raise an army, become a lord!