I am a medical geneticist.
My understanding, to become board eligible for med. gen. currently you have to do 1 of three tracks:
1) complete a "clinical residency" such as pediatrics, IM, family med, neuro, psych, etc. AFter completion you apply for a "second residency" in medical genetics. This is a 2 year program minimum. Some are arranged as 3 years I think but this isn't common. AFter that there are options to do additional training ( a year each) to sit for the subspecialty boards of cytogenetics, molecular genetics, or biochemical genetics. These traditionally have been for people who are interested in becoming lab directors. In medical genetics residency you technically function as a fellow but its called a residency, as I understand it, due to GME issues and certification. You do not have to sit for your clinical residency boards to take the genetics boards, as it is credentialed as its own boarded specialty. However, some places that hire you may want this (many don't care). This is the way I did it.
2) You could enter into a combined genetics program, such as peds/gen, OB/gen, IM/gen. These are becoming more common and these residents do both concurrently. I think programs get to decide how to arrange the scheduling but often you flip back and forth between one and the other. I personally don't see the point to this unless you know for 100% certainty that you want to go to the same program for your clinical residency and genetics residency. It narrows down your options and what you think you might want just out of medical school may not be want you want later. Most medical students have no idea about what medical genetics training programs are good and how they are different, so I would recommend personally for option #1. It also allows you to really learn your primary field and then dive into med. gen. full force because intellectually it is a demanding specialty given you see so many rare things that few other doctors have much familiarity with.
3) I believe there is still an option to enter straight into a med genetic residency right out of medical school, but few training programs have this option. I think in such circumstances it is a 4 year program but what I have heard is you do the first 2 years kind of functioning as a general intern in clinical settings and then you transition to genetics. I don't know of anyone who has done this and I have heard its not recommended.
There has been discussion about getting a "fast-track" option for mini-fellowships in genetics for phsycians in internal medicine, etc. who want to focus on just adult genetics or cancer genetics or something like that.
A few training programs have a medical genetics training option for PhDs. Most of these doctors end up doing research in labs or running cytogenetic or molecular diagnostic centers. It is different from doing a regular genetic PhD program because the focus is on human disorders and diseases with a goal toward medical care. A very few PhDs end up doing clinical work but there are fewer and fewer of those nowadays.
It is not that competitive because there are a decent number of training programs and not that many people want to go into the field at present. Once you are done, the main issue in finding a job is who is hiring because the departments/divisions are not big. You basically are looking at an academic center or university or a very, very large hospital with an academic bend. Typically a state may only have need of a handful of these doctors (more than 10 would be a lot of medical geneticists who are actively seeing patients for all but the largest states) so where you want to live may not be where jobs are available. You need to be flexible in that regard. Overall the lifestyle is good if you are into academics. Unless you are a metabolic geneticist, there are no reasons to ever have to get paged in the middle of the night. I never go to sleep next to my pager. You see cool things all day long that no one else has much familiarity with. You can play a very helpful role to patients and primary care docs in terms of education about these disorders. Some of us follow our patients for a lifetime and some patients look at us as the one doctor who truly understands their child's/their condition. That does give us a special kind of relationship that is often wanting in medicine today.
The pay typically isn't that great compared to what many medical students think a doctor "should make" but I should add that what you think is typical in medical school ends up not being always so in reality. I think 110-140K is a fair estimate in most settings but I am aware of higher and lower than this. However, this is not altogether out of line with many positions in academia to be quite honest. But keep in mind the lifestyle, never waking in the middle of the night, and fact that I see a fraction of patients that my primary care colleagues see in a day make it worth it to many. Though, my assessments are easily 1-2 hours long and sometimes more. There is often not the time pressure of most other specialties.
If you don't want to work for a university and agree to the academic life of publishing as part of your career, then this job probably isn't for you except in a few rare circumstances.
Some OB/GYNs who do genetics go into MFM and these are highly regarded positions in many centers.
Peds geneticists typically do assessment for syndromes, prenatal counseling, or assessment and management of metabolic disease (or all of the above)
Counseling/education of patients and families is a big part of our job. We work closely with genetic counselors (masters prepared position at a minimum) to do this.
Those in internal medicine often have an interest to do evaluations of adults with genetic syndromes or do things like cancer. There are relatively limited jobs available for people in this area to do this exclusively at all but the largest centers. Some IM doctors interested in common disease go into medical genetics and do epidemiologic research but have a limited clinical role because there isn't a lot we can do from a practical sense at this time to assess for the genetics of common disorders in the current routine clinical setting.
Some neurologists go on to do genetics and do "neurogenetics" which deal with neurologic disorders with a genetic basis. These physicians are usually very well regarded in childrens hospital settings because of their familiarity with many rare neurologic disorders. They often do a combination of regular neurology and genetic neurology.
I know of a few family medicine docs who went to do genetics though I am not sure how they have modeled their practice to be different from other clinical geneticists I know.
You will always have a role if you are a peds geneticist at a childrens hospital. These are much-needed and valued consultants and most childrens hospitals have at least 1.
I hope this helps. It is not an especially lucrative field but it is highly intellecutally stimulating and demanding and you see something new every day in clinic. By and large we are called to give an opinion when no one else can figure out what to do or when it is clear we have special experience on a subject. Very few geneticists actually function as the primary doctor for inpatient care and we are generally playing the role of consultant. Typically, our input is felt to be interesting and educational by our colleagues, if not helpful.
Oh, that's the other thing. Don't go into this field unless you feel very comfortable not knowing the answer or admitting you are wrong. In the best of circumstances, we may come to the correct diagnosis only 50% of the time and that even with working your butt off trying to come up with one. And you always have to consider the fact you might be wrong unless there is a way you can prove it, so you need to reconsider the diagnosis always....