Healers receive training, possibly at St. Mungo's, but we don't know the specifics.
We know there aren't many (if any) institutions of higher magical education, so there doesn't seem to be a direct counterpart to Muggle universities. It seems that Healers do go through some form of training, as there are reasonably strict academic requirements (not unlike medicine courses):
“Well, I don't fancy Healing,” said Ron on the last evening of the holidays. He was immersed in a leaflet that carried the crossed bone-and-wand emblem of St. Mungo's on its front. “It says here you need at least an E at N.E.W.T. level in Potions, Herbology, Transfiguration, Charms, and Defence Against the Dark Arts. I mean… blimey… Don't want much, do they?”
— Order of the Phoenix, chapter 29 (Career Advice)
The fact that St. Mungo's are handing out leaflets for a career in Healing means that they probably play a central role in training new Healers. Possibly the entirety of your Healer education takes place there, but this isn't confirmed.
One aspect of Muggle education that's definitely mimiced in the magical world is shadowing: students following around a qualified professional while they do their rounds:
They climbed a flight of stairs and entered the “Creature Induced Injuries” corridor, where the second door on the right bore the words, “‘DANGEROUS’ DAI LLEWELLYN WARD: SERIOUS BITES.” Underneath this was a card in a brass holder on which had been handwritten Healer-in-Charge: Hippocrates Smethwych, Trainee Healer: Augustus Pye.
— Order of the Phoenix, chapter 22 (St. Mungo's Hospital for Magical Maladies and Injuries)
The models of medical research and funding in the magical world are probably too narrow and specific to be discussed in canon. I can't recall any canon in which details are mentioned.