The short answer is, not a lot, because they've already done everything.
According to the Q known as Quinn, at the beginning of what he describes as the "New Era", the Q reveled in their newfound powers and explored the entire universe. They evidently brought back their discoveries to the continuum where they would be discussed and digested at length.
Unfortunately, an infinity of life has taken its toll on the Q. Now mostly they just sit around, going through the motions, ensuring that lesser races follow their allotted paths and don't get out of place. Occasionally one of their number goes a bit doolally and they're forced to intervene but this is apparently quite a rare occurrence:
Q2: And it was wonderful. For a while. At the beginning of the new
era, life as a Q was one continuous dialogue of discovery and issues
and humor from all over the universe... but look at them now... listen
to their dialogue now...
TUVOK: I am afraid I cannot hear any.
Q2: Because it's all been said. Everyone has seen everything, heard
everything. They haven't had to speak to each other for ten millennia.
There's nothing left to say. - Voy: Death Wish
Much of the Q's "purpose" (or at least their actions) are simply unfathomable to us lesser mortals. The metaphor that Quinn uses is that he's spent some time as a scarecrow, just because he hadn't done it before. Whether this relates to some time period spent guarding the galaxy or in self-enforced solitude isn't clear. What is clear is that the Q are bored, bored, bored. They lack purpose and direction as a people and are essentially in terminal decline, or would be if they weren't immortal:
Q2: (acknowledges) I traveled the road many times. Sat on the porch ... played the games... been the dog... everything. I was even
the scarecrow for a while.
Q2: Because I hadn't done it.
Q1: Oh, we've all done the scarecrow, big deal. - Voy: Death Wish