I'm looking for a fully functional (manual) transmission / gearbox for my Technics motorized vehicle. How can I build this?
Funny that you are asking because I just rebuilt my old 4x4 OffRoader (8466) with my son. it's a 4 wheels drive, V8 with a functional 5 speed gearbox (including reverse).
and in the truck (with the gearbox cover):
Looking at it in action is really good to understand the basics of a car's transmission.
I made a compact 6-speed, which you can find CAD and .pdf instructions in this brickshelf folder. (Maybe one was reverse, I don't quite remember...) Please bear in mind that it wasn't intended to be motorized and might not perform well under load. However it should give you an understanding on how it can be done.
If you do not have those special gearbox transmission parts, you can do it by sliding a cross-axle with gears on, in one position it will be connected to one gear, in another position it will be connected to another gear. See this design by Sariel for clarification.