In the US--warning! different from UK!--the most common and most polite phrase is the euphemism "to get sick!"
In the US the word "sick" has a range of meanings. It can mean any kind of illness, or it can mean, specifically, vomiting. When the meaning is "vomit," this is easily made clear from context. For example, in 1992 when US president George H.W. Bush vomited at a state dinner, the ABC news story title was "President Bush Gets Sick in Japan." If someone said "Sorry we're late, our six-year old grandson got sick in the car on the way here," you would understand that this mean "motion sickness" (nausea and vomiting). If someone says "I got sick three times," it is clear that "sick" means "vomit."
"To throw up" is a normal and specific expression. If I were talking to a friend, a work colleague, or even my boss, and I chose to be direct about it, that's the phrase I would use. "I was sick as a dog; I throwing up all night."
"To vomit" is slightly formal, and would be used in a "distanced" sort of way, perhaps when talking to a doctor.
"Puke" is interesting. In theory it is a perfectly ordinary English word. It is _not_ slang! Shakespeare uses it in a famous speech, "the Seven Ages of Man," in which the earliest stage is "the infant, mewling and puking in the nurse's arms." In reality, though, it is considered a strong or offensive word. It is similar to "pee."
If you say someone "vomited" you are pretending to be neutral. You are pretending that this is not a disgusting act. If you say someone "puked" you are saying that you find it disgusting, even contemptible. It would be disrespectful to say "George H. W. Bush puked." A more likely context would be "So-and-so got so drunk he puked on the sidewalk."