Professionals start with breaking down the design into smaller blocks that can be designed somewhat independently. This is very similar to how software people break a problem into function, modules, classes, etc. Of course, it is mostly experience that helps us subdivide the design into blocks that make sense.
With a Vacuum Robot, I would break up the design into the following blocks: Microcontroller, sensors, power, motor control, communication, and motors. Additionally, there are the following non-EE blocks: Chassis, dust collection, microcontroller firmware, wheel drive, possibly PC/Mac software, packaging, manufacturing, sales, technical support, test and validation, etc.
I came up with those blocks based on experience and a guess at what the requirements would be. Of course, if I had read requirements I might change the blocks. But odds are that those blocks are 95% correct.
Next, I would figure out the electronic and non-electronic parts required for each of the blocks. Of course there are dependencies to be dealt with. For example, the motors depend on the dust collection system and the wheel drive system. Motor control and drivers are dependent on the motors. Power is dependent on everything. I should at least have a very good idea about what the major parts are before starting to draw out schematics. I would also work with the chassis designer before starting on the PCB layout.
I would not use a breadboard at all. Breadboards are of very limited usefulness for professionals for several reasons. Circuits are often too complex for breadboards, either in size, speed, or other electrical properties. The parts that I would use in a robot are often incompatible with breadboards (surface mount parts). And breadboards are too delicate, too error prone, and too labor intensive. Professionals almost always go straight to designing a PCB for the product-- even for proof of concept designs.
There is nothing special about what comes after this. I just work on each of the blocks until I'm done. There is a lot of iterations. As one block changes, I might have to change another block. For example, changing to a bigger motor might change the size of the battery. I also go back and forth with the other engineers (software, mechanical, etc.) making our respective parts work well together.