Yeah, both previous people are right. I work in Critical Care, and I've worked with gunshot wounds, but I would need caliber of bullet, angle, and what it is you want to happen exactly. There are so many things that
could happen. The bullet could lodge in a bone, or fracture bones and pierce both lungs, or ricochet around in there and do major damage. The last gunshot wound I helped treat, he was shot in the hip, and though the bullet went straight through, the caliber of bullet was big enough it meant the hip bone was shattered like a dropped piece of glass. So many things could happen.
Just general info if you want something clean:
He would be rushed to emergency surgery to fix whatever internal damage you want to happen. Stop the hemorrhaging in the lungs if the bullet went in there, remove the bullet if it's in there, fix the bones, stitch up everything (if it can be, sometimes it's best to leave the wound open or partially open). Surgery often takes hours if it's severe, which it is. He'll be emergently intubated and on paralytics. When all that is done, he'll be transferred, often still intubated, to the critical care unit. He'll be kept intubated and on paralytics, blood pressures meds, and heavy pain meds until everything stabilizes, and that depends on how bad you want him hurt. Some people come off the intubation after a day or two (depends how fast they can pass a spontaneous breathing trail to breath through the tube on their own and how fast they can wake up when all the sedation is taken away). They might give him a PCA, which is a morphine pain pump that lets him push the button to control his own pain meds for a while (like an epidural pump for childbirth) or give him morphine or dilaudid every 4 hours. He would have a central line or a PICC line rather than a standard IV (or at least that would be procedure at our CCU).
For the dressings, likely he would have a wet to dry dressing changed every day, which would involve packing the wound with gauze wet with a saline solution pushed into the wound with a sterile, long qtip made for the purpose. Then we would put what is called an ABD pad over that and tape it down.
Generally, if there are no complications at all, he might be with us a week or longer before going to a regular room on a different floor. When they get ready for him to leave the hospital, they'll transfer him to a psych hospital like we do with all of our suicide patients and he'll be there are long as they deem to keep him. Just some random things are that we would bath him every day (washcloth bath in the bed performed by a patient care tech), he would be turned every two hours with pillows alternated under either side and laying on his back, they monitor his bodily fluid outputs every two hours after surgery (this is actually really important) so someone is pretty much constantly in the room looking at a catheter/suction container (he'd probably have a little nasal tube if he ends up intubated longer than a few days to suction if he has too much stuff coming into his throat or to give him tube feeding)/tube feed amounts/how much his IVs have given him etc or to ask him how he feels because there are hourly assessments on CCU patients (at least in our unit). CCU patients have non-stop telemetry monitoring (unlike what you see in movies) so they stay hooked up to a 6 lead heart monitor, blood pressure cuff (on a 30 minute cycle usually, if not every 15 minutes if his pressures are low or high), and O2 probe all the time and if they take any of those off (like the do in the movies all the time) the nurses at the desk will know about it immediately because have alarms that go off to tell us if you've taken them off, and we will come down there to put them back on you because those alarms are loud and very annoying. Our rooms in the CCU also have cameras, so we can see you at all times, so we can see if you are just taking things off or if you're in legit trouble, so we know to rush down there cause you're dying or walk down there to get onto you about taking off your O2 probe for the 15th time. I also just watched a show where someone unhooked a person's respirator tube (the one going down their throat that you get right after surgery like your character would have) and just watch them suffocate to death, and that is impossible. Those things have loud, irritating, and very obvious alarms that go off when they aren't even unplugged (they go off when you cough with one in or if the breathing isn't even exactly even, and the alarm for when they are unplugged if even louder and more angry than the one that happens when you're coughing), so there's that info for you too.
I hope I've given you a lot of info to work with. If you have any other questions I might be able to help with, or if you think of a more specific set of circumstances for your gunshot wound, I'd be happy to come back.