Designing a single-board computer and getting it manufactured
1) Buy a microcontroller development kit to start. Maybe an STMicro STM32F4Discovery because they are around $15 and pretty powerful. Spend some time learning how to program it, wire some stuff to the pins and learn how to work with peripherals like GPIOS, PWMs and ADCs. You will need to pick up some electronics background as you go, just to understand what these things are, what they do, and how to use them correctly.

2) Read the schematics for the board, in the case of the STM32F4Discovery, they are here. You will need to fill in more of your electronics background to understand the schematics.

3) Learn to use an EDA tool so you can do your circuit design and board layout. Eagle CAD is a decent choice for hobbyists. There is a great set of tutorials for Eagle here.

4) Use this knowledge to create your own system schematic and board layout, leaving out connections for the stuff you don't need. Basically use your development kit's schematic as a template and remove all the stuff you won't need for whatever your project is.

5) Submit the layout to a board house for manufacture. This is surprisingly quick and inexpensive.

6) Solder parts to the board.

7) See if it works.