in switzerland there is a "diplomat's school" which you can apply to after graduating with a top masters in any discipline. there is a strict entry test and if you pass, you get two years of education - fully paid - before you go on duty as junior diplomat. you will then be sent around the world, never staying anywhere for longer than 2-3 years, for a while. if you then manage to become a senior diplomat you can stay in one country a bit longer.
many diplomats these days work on trade agreements and supranational policy stuff. they don't go to one country but rather form a part of large negotiation teams. you often get people with any kind of economic/legal background in these teams, rather than "trained" diplomats.
there is also the track of the UN - simply apply for an open job there.