High school is 4 years. You can't choose your classes freely, but different schools have different curriculums, some are specialized (electrotechnical school, medical school...), some are general (called "gymnazium"). Specialized schools have few study programs that have somewhat different classes (e.g. for electrotechnical school you have telecomunications, computers, energetics, multimedia...). You pick one when applying for highschool, before doing your entrance examination. Some study programs have stricter requirements, based on grades from elementary school (8 years, btw) and results of entrance examination. Once in highschool, there is (generally) no changing of study programs. Classes are fixed, as predetermined for your study program, and there are no credits, you must complete all classes.

After highschool, you can apply to a "higher" school, which encompasses three years, or to an university (4 years). I'll describe how it works at the Belgrade university since that's where I study, and it's also the largest one from the few we have. Anyway, the university is composed out of a number of faculties. Each faculty is completely independent from each other, and their facilities are dispersed all over the city. In order to be accepted into the faculty of your choice, your grades and entrance examination result are used (similar to highschool).

Faculties probably vary, but generally it is somewhat similar to highschool. There are study programs, and you pick one. At my faculty, the first year was identical for all students, and then you had to pick a study program in second year. After that, you had predetermined classes for your study program. There are some alternative classes you can choose (e.g. linear statistics vs decision theory), and some selective classes (4 of them in fourth year at my faculty) where you can pick them out of a few dozen available classes. I attended the Faculty of Organizational sciences which has study programs for management, information systems and technologies (my department), quality control and operational management.

How this all works "under the cover", and as a prelude to the PhD story (btw, did I mention I am a PhD student? I am a PhD student), is that there is a number of departments at each faculty. Each department has its own professors and assistants, and it's own set of classes. Since classes are mostly fixed, you'll eventually have some classes with every department. If you want to continue education after undergraduate studies, it's a good idea to pick selective classes with a department you're interested in. Also, in order to complete the undergraduate studies, you must write and defend a thesis. You go to a department of your choice and suggest or ask them for a topic you could do.

After undergraduate stduies come the master studies. Again, you generally do an entrance exam, although in some cases you can skip (if you completed your studies at the same faculty in four years with good grades). Master is neccessary if you want to do PhD and usually lasts one, perhaps two years at some faculties. Master studies can have similar study programs to undergraduate studies, but with additional, more specialized study programs offered by some departments or combinations of departments.

In my experience, master is rather crappy. At best, it's like another year of undergraduate studies, at worst, it's much easier and just a lightweight repeat of some undergraduate classes. Why is it so? Well, my faculty is somewhat well regarded, and students that completed undergraduate studies at other faculties (especially private ones) come over to get a masters diploma to gain some credibility. This is called "diploma washing". The result is that master classes are terribly low-level in order to allow students that don't know left from right to come over and pay tutition.

PhD studies are somewhat similar to master studies in regards to departments and available study programs. When applying, your undergraduate&master studies grades are used for ranking, and additional points are awarded for published papers (in national and international conferences, journals and monographies). My facuty takes in about 25-30 PhD students every year (for comparison, it also takes about ~900 undergraduate students).

Top ~4 students could study for free, at least until the year I was admitted :/, currently I think there are no free studies at all. Other faculties might differ, but most students likely pay. A year at my faculty costs about 240.000 dinars which is, according to google, ~2200 dollars. For comparison, average monthly pay seems to be around 400 dollars. PhD studies are three years long (more on that below), so this is very expensive. There are no student loans like I often read people complaining about on Reddit, so most PhD students are already employed (wouldn't be able to afford it otherwise), with their studies sometimes financed by their employer. Some students get free studies if they get a scolarship of the ministry of education. This scolarship requires you to work on one of the available "projects" and you get payed about ~300 dollars a month in addition to having free studies, which is nice. You must be officially unemployed to be eligible, though. Only a small number of PhD students get this scholarship. The projects are usually held by various departments, so you just apply to the project held by your department.

PhD studies are, at minimum, 3 years long. That is to say there are three distinct "year-units" of study, but I don't think most people manage to actually finish in three years. I don't know firsthand how is it after finsihing the PhD studies - I'm currently at my third year. My area of study is e-education and mobile education mainly, so there isn't much lab work included. I have to admit that I also don't feel very science-y (considering that your question is about science). I've done only one research during my three years, but I was lucky enough to get published in a journal on SCI list, which is a requirement for doing my thesis. I've done some practical work developing a mobile application, but most of my work is theoretical about applying various e-learning concepts in mobile environments. I believe a lot of other students do even less practical work, although I see some interesting stuff from time to time.

I'm not sure what a post-doctoral fellowship is, but I believe there is no such thing here. If you're on really good terms with your department, if you worked as a student-assistant (for free, of course), there is a possibility that they'll ask you if you want to stay as an assistant. Other than that, you're on your own. I don't know how much a PhD is actually valued here by employers. Might be weird since I'm a PhD student, but I posses a particular practical set of skills in web and Android development, so I probably won't use my diploma for much. I'm mostly going to take it just because I can, and then I'll put in on my wall or something and open a new account on Reddit called dr.knezmilos13.

If you're interested in anything else, ask, I'll be happy to answer.

And now I'll go and lie down because I literally got tired typing this thing out.