Regarding music, Italian, French and German and Latin and Spanish play roles, with Italian given the dominant place. For example, musical directions are usually given in Italian: allegro (fast), adagio (slowly), adante (moderately slowly), decrecendo (gradually softer), a capella (unaccompanied), etc.
Some German words used in music are am steg, a term meaning to play a bowed instrument at the bridge. Another German word used is ausdrucksvoll, meaning to play expressively.
Among the French words are assez (rather), cedez (yield, give way) en retenant (slowing), and en pressant (hurrying forward)
Latin: quasi (somewhat) as in quasi-recitative of opera; fuga used with fugue for a short piece that gives way to another.
In government--laissez faire, e.g. a law term such as venue--and especially in the military, French terms are used. In fact, the military uses the Napoleonic code. Terms such as reconnaissance, sortie, rendez-vous are French. All ranks, for example, are in French. (corporal, sergeant, colonel, general, etc.)
Of course, most law terms are Latin (e.g. sine qua non, quid pro quo, pro bono, etc.)
Quite a comprehensive question there! You might want to think about narrowing it down somewhat to focus on specific areas. However, two languages that have had massive impact in the area of medicine and religion are Greek and Latin. One thing you might find it really useful and interesting to do is look up some key terms using a dictionary that gives the etymology (or the roots of words - where they have come from) of each word, and you will see how much of an impact Greek and Latin have had on the language that we speak today. However, just consider a few examples:
theology - comes from two Greek words, theos (God) and logos (word).
leukaemia - is derived from the Greek word, leukos (white).
These will get you started - now think of some more key terms that belong to these groups and look them up to find their derivations. Certainly the study of Greek and Latin is recommended! It is such a shame it is not offered widely nowadays.