The status of scientific education in the 19th century is a very complicated mess, especially for the fact that every country worked different than the others. I know of no book or article that tackles this problem in general, so I'll attempt an answer based on various readings, majorly biographies.
1) Basic Education in the 19th century
To understand what happened in the universities one must understand the state of primary and secondary education in the 19th century. Until the 1870s there were few laws regarding education. Most of the population were illiterate. Wealthier people would send their kids to primary education in order to learn how to read, and secondary education would consist chiefly of Latin, Greek, Classic Authors and Religion. While I'm not certain, it is possible that schools would teach the very basics of euclidean geometry. What is certain is that more advanced mathematical topics like analytical geometry were left to higher education. The sciences were not seen as relevant to an education which existed to create an intellectual elite. So it is not that what we see today in schools was not taught, but no science at all. This situation was somewhat maintained up to the turn of the 20th century to various degrees. Heisenberg's autobiography (W. Heisenberg, Physics and Beyond) mentions that he studied the concept of atoms at school where the book illustrated a chemical bond as the atoms having hooks. In the UK, Dirac's biography (G. Farmelo, The Strangest Man: The Hidden Life of Paul Dirac) says he did study science and modern languages at school, but that was unusual because he was lower class and attended technical college, while more wealthy people still went to grammar schools which focused in Classics. Widespread science teaching in secondary schools is a cold war phenomena where countries put a lot of effort in creating a lot of engineers and other science professions, see arms race and space race.
2) The State of Physics in the 19th century
What we now study in undegrad physics were not seen as unified field at that time. Mechanics and Optics (as well as astronomy) were seen as part of mathematics, and the experimental sciences were grouped as natural philosophy. The term "physicist", meaning someone who used deductive reasoning in experimental sciences was only coined in 1840 by William Whewell, and took some time to get going (which you can see using google ngram). So if you wanted to study mechanics you would get a maths degree, and if you cared for experiments you would study natural sciences (the natural sciences degree at Cambridge started in 1851, although I think it may have precedents in Germany). Getting both together is a late 19th/early 20th century phenomena. Plank's autobiography (M. Planck, Scientific Autobiography and Other Papers) mentions that the concept of theoretical physics was starting to be discussed by the time he took his PhD in the 1880s, majorly because of problems involving thermodynamics, like the blackbody spectrum. By the way, this distinction of physics in two parts is the reason why theoretical physics is still in the mathematics department in the UK and former colonies (and I suspect in the Soviet Union)
3) The State of Physics Education in the 19th Century
Your question is mostly focused on what did people in fact study. Here it gets messier because each country had it's own system. If we take Cambridge as representative of the UK, there the undergrad education was based in the Tripos system. Basically you hanged around the university for 3-4 years taking random classes and studying with a private tutor. At the end of your time you took the Tripos exam, a single test lasting days, and if you passed you got a degree. There was no PhD studies per se, after the Tripos people that showed promise might be offered a fellowship, a stipend by one of the colleges, to teach and tutor. If they did research it was usually to be connected to Royal Institutions of some science of another.
Germany is the birthplace of the undergrad/PhD system in the early 1800s. During undergrad you were expected to attend whichever classes you wanted. Matter of fact a lot of professor did not receive pay from the university, but rather they held a license to teach and were paid according to the number of students attending the classes they offered. The PhD was when your worked under a supervisor and had to produce a dissertation to be examined.
As far as I understand France was the place where you had rigid curriculum. The system of Grande Ecoles and Ecoles Normales had predetermined classes and was based very strongly on the lecture. France did not have PhDs until the late 19th century, nor the idea of research university (although of course a lot of french scientists did research, mainly in the british style of connection to other institutions).
Bottomline, during most of the 19th century the universities did not have rigid curriculum, so to answer what people did study you have to dig the examinations they had to pass. It is noteworthy that in the 1850s the Stokes' Theorem appeared in the examination for the Smith's Prize, an optional part of the math Tripos at Cambridge that evaluated original work instead of previous study, as a question to generalize Green's Theorem to 3 dimension, from what we surmise that it was not explicitly taught. That may give you some idea of at least the math level. In any case most classes were made by the professors from his personal notes, so say taking a ODEs class from professor A might be very different from taking the class with the same name from professor B.
[as an addendum, the lack of widespread PhD system is the reason why the Math Genealogy Project has to make some educated guesses when establishing supervisor relations before the 20th century for a lot of cases]
4) The State of Thermodynamics Education in the 19th Century
Since I think the answer was less than satisfactory, I'll try to point out that the idea of thermodynamics as based on the first and second laws, and the understanding that thermodynamical energy was just another form of energy took a long time to develop. Planck's biography (the same as before) mentions that he struggled in his early years to cobble together what was known into a unified frame, and that only later he discovered that Gibbs had done the same in the US (both 1870s), while Silvanus Thompson's Elementary Lessons in Electricity and Magnetism (1890) says it was still discussed if electric energy was an energy like the others, so it is not surprising that Kelvin was learning from original papers about the first law of thermodynamics (Joule did the experiment, Helmholtz the theory).
As for Fourier's theory of heat conduction, is good to note that the Heat Equation is a poor empirical description of heat transport in most situations (Joel Lebowitz has a paper on the Heat Equation where he discusses that only in a copule of material, with low temperature differences one can actually model the transport as in Fourier's work). The big news here was Fourier suggesting that all functions could be represented by a trigonometric series in a given interval. By that time issues of convergence were poorly understood, I guess up until Weierstrass in the late 19th century, so the polemic seems to be normal.
[I've tried to remember things as best as I could from all the biographies and books on the history of mathematics and physics, but I'm sure there are a couple of errors creeping in, so take everything with lots of grains of salt. Hopefully the comments will be filled with corrections and I'll update accordingly. I do apologize for any mistake that might mislead someone.]
