Would they have to go to university and take a similar route to nowadays?
I’ve heard horror stories of doctors from these times that sounded incredibly inept at their job.
Sort of. The medical educational system was far more heterogeneous prior to the 20th century (in the United States, prior to the publication of the Flexner report). Let me focus on just one medical school, my alma mater, Tulane University, which was founded in 1834 as the Medical College of Louisiana. The Matas Library there has a fascinating reprint of its early history (http://www.tulane.edu/~matas/historical/SpecColl/pamphlets/Fossier1935.pdf).
So the charter of the school speaks of the importance of having affordable, English-language (as you can imagine, French is still commonly spoken at this time in New Orleans) medical education because of the proliferation of "Quack Doctors" throughout the state. They also explicitly design the school to focus on teaching at the bedside -- to take place at the Charity Hospital. This in and of itself is quite forward thinking at the time, and shows the influence of the Paris (of French) School on American physicians, which stressed bedside teaching and autopsies.
There were initially just five lecturers:
Other than bedside teaching, there were daily anatomy lectures, and twice-weekly "chemistry" lectures. The school didn't have a physical location; many lectures took place in a local Unitarian church, as well as in Charity Hospital.
A couple things to note about the quality of the education here. None of these "professors" were academics in any meaningful sense. They were very young -- four of them were only 26, and there was quite a dust-up since they were "self-appointed;" they had no academic bona-fides. Secondly, the school was not chartered; Dr. Hunt (the anatomy lecturer) had entrepreneurialy decided that Louisiana needed a medical college and set about starting one (a state charter would come later). And there was no examination system for the students -- they essentially attended a predetermined number of lectures, and were granted a medical degree. This is also an undergraduate medical degree -- the men attending these lectures would, for the most part, not have had a university education.
Now, the Medical College of Louisiana (renamed to Tulane post-Civil War after Paul Tulane donated a large sum to keep it afloat) certainly changed over its history (especially at the end of the 19th century) into something more akin to a modern medical university (under the influence of the German School, which advocated a unification between the basic sciences and medical education in a university setting) pre-Flexner. But up until the Flexner report was published in 1910, many medical college throughout the United States would have been similarly organized, with no uniformity to their curricula (and in fact, a fair amount of pseudoscience was taught in them). This is also the period in which medicine in the United States becomes a post-graduate study (in much of the rest of the world, it's still an undergraduate degree).
So this education is rather heterogeneous, but it's still recognizable to today. The biggest difference is probably the absence of post-graduate training, what we call an internship and residency. Traditionally, physicians and surgeons would have apprenticed by experienced practitioners, learning the tools of the trade as it were. Again, you can imagine the heterogeneity. This was not strictly required; many of the graduates of the Louisiana Medical College would have gone and hung up a shingle. Who would prevent them otherwise? A big innovation in the United States towards the end of the 19th century was the development of "residency," a post-graduate training period of intense inpatient study and care (hence the name "resident"; they lived in the hospital). William Osler is probably most remembered for developing the medical residency at Hopkins which was emulated throughout the country (and, in varying degrees, the world). I can tell you from my own experience that it is residency where you learn to become a doctor, not medical school -- much of what we learn there is not clinically useful.
So a talented young doctor at the Louisiana Medical College would likely complete an apprenticeship after, hopefully with someone good -- but it wasn't formalized, and there were no guarantees.
I hope that answers your question!
I do want to briefly address your second point, of horror stories and doctors being "inept" at their job. In this period, the quality controls we have today were not in place, so there were in fact a lot of "inept" doctors. But the period you're interested in, the 18th and 19th century, is my favorite period of medical history -- there's an explosion in medical knowledge, but the tools of medical epistemology (how we know things) are only slowly developed and utilized. Doctors in this early period would not have had any good way of knowing whether their treatments work; doctors in this later period struggled with some of these novel methods that argued against their experience. This struggle goes on to this day. Anyway, I could write about that topic forever, so I'm going to call it quits now. Hope this was helpful.