The Philosophy of Science

You want some guidance on pursuing study of Philosophy of Science (I take it you mean natural science rather than social science or economic science). Such a subject is only part of any degree course. It was part of my B.A., (Maths and Philosophy) and my Ph.D. qualifying exam (Logic and the Methodology of Science) included it. There are good reasons why it needs to be studied in tandem with other things. For example, it has proven very difficult to distinguish science as an epistemological activity from other such activity. Attempts to identify a distinctive “scientific method” that reliably leads to knowledge have so far generally failed (there was a lot more hope 80 years ago than there is nowadays). It makes sense to study Phil Sci along with a science (for example you can study Maths & Philosophy, or Physics & Philosophy, at Oxford; Medicine & Philosophy makes sense), or along with the History of Science (much Phil Sci indeed looks at the history), or as part of Philosophy more generally.  

I am also a strong advocate of combining study of such disciplines with developing practical skills - the accompanying science in the <science>&Philosophy courses above; economics or finance for business; accounting; medicine & health; IT; engineering of some sort. Too many young people finish a 3-year or 4-year degree course without having obtained any specific skill that they can offer in exchange for a salary. That is OK for some, but manifestly not for all.

Thanks. If you look at these programs mentioned in the article, there are programs that emphasize a natural or social science to study in the context of the history and philosophy of science. 

Yes, there is a long tradition in the US of binding philosophy of science to history of science. That fulfils at the graduate level (what I intended to be) my criterion, but not ideally, because there are only academic outlets; you can get an academic job if you're lucky (although there aren't many) but you can't just go off and work at Argonne.  There is also the STS tradition, from the 1970's onwards, which I don't think you hinted at, so maybe you're not that interested in it; the people who look at science as a social phenomenon and scientific knowledge as the result of social agreement, close to an American Pragmatist take. And arising out of that the “science wars” which attracted many philosophers who were not previously into Phil Sci. People interested in either of those have a wider choice of program than those you originally listed; but, again, they typically lead to back to academia, not to immersion in scientific or science-related activity.  I also note my program, Logic and the Methodology of Science (aka “Tarski's Group”) is not listed. 

I still wish to emphasise that combining with something useful outside academia is a very good move. I remember a post-doc physicist pal in Berkeley in the late 1970's bemoaning the fact that there had been one tenure-track physics job in his field offered in all of the US in that year. He was more of the math type. My metal-bending physics colleague set up a consulting firm which ran largely on USG research contracts and he ended up going to all sorts of fun places.

A possibly-fruitful new line, and not well represented in the listed programs, is Philosophy and/of Medicine, which I suspect is best pursued along with medical training (and therefore not in the US, where medical training is post-grad, all-consuming and very expensive). Medicine is full of things which I like to call ontologically malleable; many things are there or they are not there, depending. When you break your arm, it is clear what is wrong, but what about Encephalitis lethargica? Or ME/CFS? An advantage of such study is that, if you are not a medic by the time you graduate, you can at least go work for an insurance company :-)