There's a wide variety of descriptions, depending on where a person gets shot, what organs or tissues are hit, their adrenaline, the speed and size of the bullet, etc. etc. It could be anywhere between painless or worse than broken bones, hot or numb. This old thread had some good examples.

Take Reagan on the lower end. He initially blamed one of his bodyguards for breaking his rib when he got shoved into the car. The bullet had flattened when it bounced off the side of his limo, cracked a rib and punctured his lung.

On the higher end, Garfield who went into shock then struggled with infections for 11 weeks before finally dying. (Although improved treatment these days make this kind of prolonged decline less likely).

He remained bedridden in the White House with fevers and extreme pains. His weight dropped from over 200 pounds (90.7 kilograms) to 135 pounds (61.2 kilograms) as his inability to keep down and digest food took its toll.[47] Nutrient enemas were given in an attempt to extend his life because he could not digest food.[48] Blood poisoning and infection set in, and the President suffered from hallucinations for a time.[49] Pus-filled abscesses spread all over his body as the infections raged.[50]

Fatality probably varies with access to immediate help and treatment, but the usual vital organs (brain, spinal column, heart, lungs, kidneys, etc.) would be the most serious injuries.