What objects in the solar system other than Earth have water and oxygen?

Water, although not liquid water, is reasonably common in the solar system. Earth's water, for example, is reckoned to have been brought here by early collisions with comets. Oxygen is another matter. Free oxygen as we know it here on earth is in fact a form of pollution created by plants. The rest of the solar system does not have plants, does not have photosynthesis, and therefore does not have the by-product pollutant, gaseous oxygen. Oxygen elsewhere is bound mainly to carbon as carbon dioxide in the atmosphere, and to silicon as silica in rocks.

Water is almost everywhere. So is oxygen. Water is one of the most common molecules in the universe. It's nothing more than hydrogen (the most common element in the universe) and oxygen combined. Oxygen by the way is the 3rd most common element in the universe. We've found water on the Moon, on Mars, on asteroids, comets, Jovian moons, in nebulae, etc. Of course it's frozen, but it's still water. It's even believed that Jupiter's moon Europa probably has a liquid water ocean underneath the few miles of water ice that covers its surface. It's suspected that this ocean might contain more liquid water than all of Earth's oceans combined. Oxygen is also very, very, very common. Oxygen is not stable as a gas so doesn't exist in that form much. We only have so much oxygen as a gas here on Earth because we have trillions of plants breathing it out 24/7. Still most oxygen on Earth is found bound up in rocks. Still it exists in the atmospheres of Venus and Mars in great quantities, bound up in carbon dioxide (there's 2 oxygen atoms in each CO2 molecule remember?). However, gaseous oxygen atmospheres do exist on Jupiter's moon Ganymede and Saturn's moon Dione that I can recall off hand.

Hydrogen and Oxygen (water is H2O) has been found on the Moon and Mars, it is thought there is lots of water on comets. We can take the CO2 in the Martian atmosphere and turn it into O2 with school boy level biology (you need some soil, some plants and some sunlight).

As far as natrually occurring liquid water and diatomic (breathable) oxygen gas is concerned Earth is the only body in our solar system where humans can live without artificial life support.

Water (the molecule) seems to be everywhere. Liquid water is rare, especially in the form of oceans.

Free oxygen is extremely rare, as it tends to react with any other compound. For example, on Mars, all the oxygen that ever existed on the surface is tied up in "rust" molecules on the surface (giving the planet its red color).

Earth has free oxygen because plants keep replenishing it.

Europa (satellite of Jupiter) could have liquid water under its crust.

Triton (captured satellite of Neptune) has a mantle that is probably mostly water, still in a liquid form (kept that way by the tidal braking energy that was released with Triton was captured).

Mercury COULD have some underground water (mixed in with its "soil") under ground at its north pole. The Moon could have some water under ground (also mixed in with the "soil").

Mar's current atmospheric temperature is not suitable to contain water. But scientists proved that water once roamed the face of Mars.

They have discovered Mars once had bodies of water

almost anything in orbit around the sun. getting to that object, and extracting what you need is still sci fi; but ultimately doable.