The applications process is not so different from regular physics. The background expectations are nearly identical. The curricula are different but they have to be. No one is applying field theory to medicine, nor are they likely to be any time soon.
That said, the curriculum consists of classes on the interactions of radiation with matter, radiotherapy, and the physics behind all kinds of diagnostic imaging. You will likely encounter specialized math classes as medical physicists need mathematical knowledge that differs in many ways from what traditional physics programs offer. It is usually expected that students complete these courses within the first year at which time they are expected to pass a qualifying exam (Ph.d programs). Unlike regular physics programs, this test consists mostly of material from the first year graduate courses. The classes you take after that will vary from program to program, but usually they cover the more specialized aspects of biology, health physics, radiotherapy, and medical image processing.
If you want to move straight into a Ph.d. program, apply to one. Unless you cannot get into one directly there is no need to apply to masters programs first. Two year masters programs are usually meant for those people who seek to become clinical medical physicists without having any research responsibilities. Clinical medical physicists with Ph.d.'s that work in research hospitals are expected to publish and participate in research, but their duties are mostly clinical (60-70%). It's best to make sure the program you apply to is CAMPEP accredited because that will make it easier for you to move into clinical work and to obtain a residency. If you plan on working in the field of diagnostic imaging this won't matter too much for you but it's best to be safe since plans can change. You can find a list of CAMPEP accredited schools on the web.