At the beginning there was the Wherner Von Braun team. They were german scientists & engineers really enthusiast for rocketry since before the war (the man that is considered to have "created" rocketry around 1890-1900 is [ame="http://fr.wikipedia.org/wiki/Constantin_Tsiolkovski"]Constantin Tsiolkovsky[/ame], a russian primary school teacher), and that worked for the Nazis during the WWII (were they Nazis or not, that's a debate for the Basement). Anyways they were given US citizenship provided they worked on an ICBM program (Thor, Redstone, Atlas rockets...).
So at first you had the [ame="http://en.wikipedia.org/wiki/National_Advisory_Committee_for_Aeronautics"]NACA[/ame] since the first Goddard works (US father of rocketry). Then, after the WWII, Wherner's team bring their knowledge about V2s (by far the most advanced rocket of the era), in order to develop an ICBM program. Then there was Sputnik (1957) and the space race (there were no solid projects for Mercury/Gemini before that)...
I guess that a lot of engineers learned "on the pile", getting precious experience from Von Braun team and developping their own knowledge and ideas. At first it was very experimental, unmanned launches were "trial & error", very far from the way it is done nowadays (computer simulations, checks & rechecks, redundant systems etc...)
So I guess the engineers were much more "generalists" back then. Nowadays, it is very specialized, each team working on a very specific aspect. And about the first rockets, it was almost "amateur rocketry scaled up", the rate of failure was high (even for the original V2).