Are there underground caverns on Moon or Mars? Are they expected to have air (vel: some gas) inside? Do we already know of any?
There are also NASA reports about entrances to martian caves here. More work is needed, but Mars looks promising for caves too.
There is evidence of caves on both the Moon and the planet Mars.
Yes, there may be subsurface features. Lava is known to a have flowed in the past on the moon (there may still be a low level of outgassing). When a fluid lava flows, it can form a lava tube. With relativly little to disturb it, these can last for hundreds of millions of years. In places the roof of the tubes have collapsed forming pits - Space.com video
There is speculation that the largest of such tubes could be exceedingly large. Very large tubes on Earth would collapse under their own weight, but in the moon's gravity they could be much larger.
A Tube would be a good place for a lunar base. It is protected from solar and cosmic radiation and micrometeorites. It would have a stable temperature. However, they would not contain air or gas in appreciable amounts.
As a setting for a story, you might imagine an exceptionally large pit, some cubic km in volume. Tunnels cut into the rock could be pressurised to allow for human activities. Rockets would leave via the centre of the pit to lunar orbit and Earth transfer.
There would be no water: making and controlling water supply would be a major issue, as would maintaining a breathable atmosphere, as sunlight would not be available for plants.