Which fields of physics should I know in order to understand how diodes and transistors work?
Basic, generic physics. You don't even need to understand or know about the cloud model for electrons. You can pick even that much up as you go along, really. You need to understand all the basic SI units, what they mean, how they are used and applied, the basic idea of dimensional analysis, etc. The first volume of Feynman's Lecture series on physics is more than enough, I'd imagine.

I'd recommend volume 1 of Feynman's lecture series on physics (volume 2 if you want to get into electromagnetism, too), a book by Jacob Millman called "Microelectronics" (nearly any edition is fine) for diode and transistor discussions that are easy to follow, a book called "Matter &amp; Interactions" with at least the 3rd edition as it is provides a very intuitive understanding of rather deep physical ideas,... 

But to be honest? Just Jacob Millman's Microelectronics, and even then only perhaps the first four or five chapters, is enough to get the basic ideas in your head.

I've found the web to be near to useless on this, though.