Welcome to the forum!
In the thread "Resurrecting a Roketa MC54 250B" I posted a link to a wiring diagram. That diagram is representative of all Chinese scooters I have seen. Get a copy for yourself.
http://scootdawg.net/showthread.php?t=54850
If you don't have any electrical tools, I posted a photo of some tools that are cheap but will work for the casual user.
Here is the way the power flows on the circuit you need: Battery positive --> Main fuse --> Ignition switch --> Chassis fuse (if equipped) --> Brake lamp switches --> starter solenoid coil windings --> start button --> ground
Since I've laid out the power path, you can see IF the brake lights work, everything to there is good. That leaves probing the solenoid leads while the brake switch is depressed for power on both while connected(don't rule out bent pins in the connectors). If that is good, a quick check of the solenoid is to ground the red w/yellow tracer wire with a jumper wire while depressing the brake with key on, if the starter engages, the solenoid is good.
If everything to here is good, then most likely you either bent pins in the right switch harness connector, or the switch is corroded (which does happen). To test the switch, power off, use DMM lowest Ohms setting & probe the red w/ yellow tracer wire to ground (green wire) and depress switch. If it stays high on reading switch is corroded most likely. Those switches can be removed, cleaned and reinstalled w/o replacement.