Galastel gave an already excellent answer about using the pain to propel the story forward.
I'll add my two cents:
What you want to avoid is showing the pain for the sake of it.
As you said, you don't want your character to be immune to pain, and adding details such as changing dressings and dealing with a wound will add realism to the story. You should, however, indulge in those themes; since the readers can tell if you're overdoing it (to the point of misery porn, maybe).
Now it's a fine line between doing it properly and doing it too much. Since you want your character able to act after the wound (as opposed to being incapacitated by it) you need to keep the pain at a manageable level. For the sake of simplicity, I'm assuming that the bullet has exited the body and there is no major internal damage.
The point is mentioning the pain when it makes sense doing so: for example, if it's a gut shot, it may be worth making the character grit his teeth when he needs to bend over, putting weight on the wound.
But when does it make sense? I'm going for a metaphor, so follow me. It'sa bit like describing the character's appearance: it makes sense doing it at the start of the novel. If your protagonist has some striking physical feature, it's only natural that you'll want to describe it in detail. But it's usually a bad decision to go over the same information, at the same level of detail, twice: you would not be giving new data to the audience, nor you will be progressing the plot.
In short, even if your character has the most beautiful blue eyes, you don't want to fill paragraphs about them halfway trough the novel. You're still allowed to remember the audience that - mentioning them as the character does other things - but you can't linger.
Describing pain should work similarly. There is a initial peak - were the character feels the most pain and you can go over it in detail as much as you want, and it will be allright, since it would be a "new" experience that both character and audience need to go through. After the peak, the pain intensity should gradually lower - and so your descriptions should shorten. You're still allowed to describe it when it's relevant and when something new happens: how does the wound "reacts" when the character walks? How when he changes dressings?
But you don't want to go over the same description everytime the wounded character takes a step. After all, if the pain is so unbearable, he shouldn't walk after all!