When we inhale air, other gases are also inhaled in addition to oxygen. What happens to those gases?

It's worth taking a quick look at the average composition of the atmosphere to answer this question.

Nitrogen: 78%

Oxygen: 21%

Argon: 0.9%

Other gases: 0.1%

We should consider the three primary gases first. Nitrogen and oxygen are both dimolecular, meaning, each one is a pair of atoms bound together. These are pretty small in comparison to the membranes of your cells, let alone your lungs, so it should be fairly easy for them to get into your blood. Argon is an inert gas, so a small amount of it should be irrelevant. A good example of what happens with an inert gas in your body is when people inhale helium; no observable effect besides making your voice squeaky, and maybe lightheadedness, but in sufficient amounts helium can actually be used as a form of euthanasia, so it's entirely viable to say that inert gases don't have to react with anything in order to harm you. However, you're not going to breathe in a deadly amount of argon anywhere in the atmosphere.

This means our other primary consideration should be the nitrogen, which is three or four times more of every breath than the oxygen we actually want. They're similar enough in size and properties that it's not like our lungs can "choose" which one to allow into the blood, and in fact nitrogen does diffuse into our blood right alongside the oxygen. However, we don't have any use for it, and so our blood lacks the ability to bind with the nitrogen and allow it to accumulate in any significant amount. It's there, it just doesn't do anything.

Likewise, the other lesser concentrations of gases tend to either be too small to have a significant effect, like the argon, or not really viable reactants with our metabolism, and therefore just sit around dissolved in our blood in relatively small concentrations, ignored by the cells.