It seems to me that space doesn't have any/much air, and if my bottle is full of air, when I open it, where does the air go?
It will diffuse into space.
Space is a near-perfect vacuum — its pressure is nearly zero and it has extremely little matter (in the empty parts, at any rate).
On the other hand, your bottle has a relatively high pressure. When you remove the barrier (by opening the cap), the air naturally flows to the region of low pressure.
Once there, it creates a localized blob of air, with low pressure — but not as low as the surrounding space. This localised blob will spread out due to the pressure difference.
Theoretically, the air particles will continue spreading indefinitely, if we neglected gravity, the fact that the universe has other stuff in it, and that matter is not continuous. Anyway, there will be an incredibly negligible increase of the pressure of the immediate area.
That's about it — the air just spreads out. Sort of like what happens when you put a drop of red food coloring in a tank of water — it first creates a localized red blob, which spreads out till the change in color is negligible.
Note that this process, in space, will be much faster than the red food coloring one. The two are similar phenomena but the mechanics are different.
The gas molecules in your bottle of air aren't just sitting still, they're moving around in random directions. From memory, the speed of oxygen and nitrogen molecules at room temperature is around 500 meters per second.
When the bottle is closed, the air molecules hit the walls and lid of the bottle and bounce back, so the air stays in the bottle. If you take the lid off, then because the gas molecules are moving around randomly, and at high speed, pretty soon most of them will have escaped through the opening.
Now, if you take the lid off the bottle here on earth there are air molecules buzzing around in the atmosphere outside the bottle, so at the same time as the air molecules in the bottle escape, air molecules from the amosphere enter the bottle. The end result is the bottle stays full of air.
If you take the lid off in space the air molecules in the bottle escape, but there aren't any air molecules in space to replace them, so the bottle will very quickly be empty. The air molecules that were in the bottle will be heading off into outer space at a few hundred meters per second.
When you take the lid off, all the molecules that would otherwise hit it escape since there is nothing to hold them back. Although the molecules are going at a typical thermal velocity of roughly 500 m/s, the mean free path of molecules in air is about 70 nm and it therefore takes some time for molecules near the bottom of the bottle to "find out" that the lid is open. The effect is a wave of decreasing pressure travelling from the lid to the bottom at about the speed of sound in the bottle.
As a professor of mine used to say "enthalpy rules!" The heat/energy will be dissipated into space and the oxygen molecules will slow down and travel until they find a new "place" to be used. There can be oxygen in space, but there is no atmosphere to hold it together to be used by humans.
It just works like a balloon here on earth. The air has overpressure (since space has effectively zero pressure) so if you open the exit it streams out until the pressure inside equals the pressure outside (i.e. 0). There is really nothing special about space in that respect.