If the electric start of a scooter doesn't work, press the electric button while pressing the left handle brake. If you hear a clicking noise, the battery is low. In this case, check the fuse. Make sure the key switch is working and there is no smell coming from the electric start. If there is a burnt smell, the electric start is probably burnt.
To troubleshoot a scooter that isn't charging, check if the fuse if working, and make sure the wires behind the fuse holder and the charging jack aren't loose.
If a scooter doesn't start, check if the fuel is getting to the carburetor. Make sure the spark plug is in good condition, and replace it if it has tears or signs of wear. Try starting the scooter with both brake levers, and spray some starting fluid into the carburetor to help the scooter start. If the carburetor dies once it runs out of the starting fluid, this means the fuel doesn't get to the carburetor. Replace the piston if it is damaged, as there may be a vacuum leak.
If the scooter is very slow, check its tire pressure. Make sure the wheels are able to rotate freely. Adjust the brakes if they're too tight, and make sure there isn't too much brake fluid.