starter and gas gauge
the 12 volt start wire from the solenoid is right, it sounds like you have no 12 volt run circuit. either at the ignition switch or the voltage resistor before the coil. when the stater engages it sends 12 volts to directly to the coil. when you let up on the switch you should have 12 volts to the resistor and about 8 volts out of the resistor.
The brown wire should have a cycling voltage on a test light. in other words it blinks on and off on a trst light. if it is steady the voltage regulator for the gauges may be shorted.