"However, how does the preburner begin running?"
Refer to this simplified schematic from the crew checklist for the Shuttle system I'm about to describe.
Head and ullage pressure in the propellant tanks is the initial impetus for propellant to flow into the preburner(s). At least in the Shuttle, the tanks were pressurized on the pad by helium supplied by the ground, after their respective vent valves closed. The O2 tank was way above the engines at the tip of the ET so there was significant head pressure too. Not so much head for the LH2, it does not have the density for it. Then when the valves in the plumbing between the tanks and the engines opened, and the engine valves did too, prop flowed into the preburner driven by tank outlet pressure. Then the electrical igniters kicked on, the preburners ignited, the turbines began to spin, etc, as you have described. (reference the Space Shuttle News Reference Manual, page 185 of the pdf)
There is info about the start sequence in the answer to this question but it does not specifically talk about what drives the flow initially.