Healthcare in US?

Can someone please explain to me how your healthcare system works in the US? Here (UK) everyone registers with a GP, you have any problems that's where you go, unless its an emergency (broken bones, car accident etc) in which case you go to A&E at your local hospital. Pretty much everything else you go to your GP and they decide if you need more tests, if you do they refer you to the hospital, you get a letter through your door with an appointment time and date at the hospital and you go. This includes things like x-rays, ultrasounds, blood tests etc.

Only asking this because i just saw a post about scheduling a colonoscopy, can you just ring and book one? Who would you ring? The only way we would get one here is if you had blood in your stools or something similar, you made an appointment with your GP and they decided you needed one in which case you would get an appointment for one, we cant just book one just to make sure everythings ok if that makes sense? Same with mammograms etc you only really get one if your over a certain age of have symptoms of something?

Random question i know, just interested in how our healthcare systems are different.

ps- i have been aving really bad pains in my right leg for about 6 months, went to my gp, who referred me for an x ray at the hospital a week later, then back to the gp a week after that for the results (which were fine.) Then got a letter through the door about a month later for an appointment at the orthopedic clinic at the hospital about 2 weeks after that. Then scheduled for scan and physio, which im still waiting for. Interested if this is roughly the same as in the US?