There are a few different fellowship options. Most programs are combined heme/onc fellowships, and consist of about 1.5-2 years of clinical time and 1 year of research. When you finish you'll be eligible to be boarded in both hematology and oncology. But some will allow you to just do a "medical oncology" fellowship, and skip the heme part. These are usually 2 year programs, with about 1 year of onc and 1 year of research.
Heme/onc fellowships aren't as competitive as GI or cardiology, but they're not "easy" to get either. My understanding is that they're considered to be somewhere in the middle.
To answer your last question, oncologists often function like primary care physicians for their oncology patients. But I'm sure that as an academician you could also serve as a medicine ward attending 1 month a year while also fulfilling clinical oncology duties.
