honestly i havnt played around with anything as high voltage as a neon sign power supplys or even flash circuits. Mostly i have done handheld battery applications of like 3v to 12v. The most impressive supply i did bumped 12V to 75V @ 1A continuous draw.
the big difference with say a flash circuit to neon supply is that a flash circuit is a pulsed power application where very high voltage and very low current are needed for a very short amount of time ~mSec. the Neon supply has to provide a high voltage and low current continuously so the designs will vary considerably.
The question about the transistor is a good one and there are many designs that utilize a high speed NPN transistor. Probably not a 3904 as that is general purpose, however there are specific transistors designed precisely for this application. Mosfets are rapidly replacing transistors for switching power supplies as they are much lower loss however they are more expensive.
Diodes are specific for switching power supplies as well, they need to be high speed and low loss. Schottky diodes are perfect candidates for the task.
You could spend the better part of 4 years in college learning about power supplies. My expertize lies in DC to DC converters for battery applications 3-24V and that was several years of designing circuits after quite a few years of electrical classes. My design process would start with choosing an IC that was appropriate for my requirements and then I would lay a circuit board to try the design. takes a lot of app notes and part sourcing as well as a fair amount of trial and error.
so I really cant help you more specifically then general information without knowing what kind of trouble you are trying to get into