I am starting with microelectronics engineering, but in fact if we were going to be strictly correct, I would have to start with theoretical physics, because much of the design of integrated circuits that are used on microchips involves working out the behavior of electrons as they are forced down narrower and narrower pathways. Today design parameters require increased miniaturization, and as the number of pathways and "gates" on a chip increase, so the necessity for narrower pathways becomes more urgent. At .01 micron design levels, the behavior of individual electrons become more erratic, and so the theoretical physics comes in.
Microelectonic designers design the chips we use in computers and in other electronic devices, the printed circuit boards that serve as a medium for the miniature components of these devices, and some of the other electronic components that are used in computers, printers, and control systems. Microelectronic engineers test and maintain these devices, and help to manufacture the chips in clean room facilities.
If you are interested in becoming a microelectronic designer or engineer, you will need to gain a Bachelor's degree in a relevant subject - it could be math, physics or electronic engineering, and will probably need a graduate degree in engineering or design. Many designers have a doctorate, using their design ideas as the basis for original research for their thesis.
This is a job for the skilled engineer with deep theoretical knowledge, but it can be exciting and rewarding. You will work in teams of engineers, and will find that this kind of work is really absorbing. It can also be financially rewarding if you are a talented engineer, and are prepared to work long and hard to create a reputation in your field.