Testing the spark ignition:
Remove the spark plug and connect the spark plug cap from the high voltage coil to it. Manually hold the spark plug with cloth onto the cylinder head and make sure the side electrode is sideways so you can see the spark gap clearly. Push the bike to see if any spark is generated. It is hard to see in full daylight so its best to do at night or inside. If no spark happens then there is a problem. If you have spark but the engine won't start then one of these problems exist:
1. Compression is being lost due to loose head nuts or non-parallel head/cylinder mating surfaces (that you can fix with sandpaper and glass).
2. There is an input air leak due to loose intake manifold connections or poor sealing where the carburetor connects to the intake manifold.
3. Crankcase compression is being lost due to bad crankshaft seals.
4. Something is really wrong with the carburetor such as a jet being clogged.
5. Someone (probably you) put the magneto on backwards so that the spark is happening at the wrong time.
Test the Stator Coil:
Use an AC voltmeter to test for 25-50 volts coming out of the stator coil as you push the bike with the spark plug removed. I prefer using an analog voltmeter such as this one:
If the voltage is there then the stator coil is OK although I would still remove the stator coils mounting screw that fastens the ground wire to the metal ground so that you can clean it for good conduction.
Test the CDI:
You can test for total circuit performance (without determining the peak output voltage which requires a special test circuit or a meter that can read peak voltage) by just holding the CDI output wires while pushing the bike. Someone may have to help you with that. Start out very slow and gradually push faster till you feel a minor shock. I prefer this basic method because the output voltage lasts for only a few thousandths of a second which most meters can't register. If you are getting shocked then suspect the high voltage coil or the spark plug cap or the spark plug as bad.
Next you can test the input and output circuits diodes. Using the diode test function it should show a .5 to .6 voltage drop across the diode under these test conditions:
Red probe on black input wire and black probe on green input wire.
Red probe on red output wire and black probe on black output wire.
You can't test for resistance at the input or output because the input is a combo of resistance and a diode, and the output is capacitance and a diode.
What normally goes bad in a CDI is one of the diodes, the SCR (black transistor with 3 legs), and the large output capacitor (the largest component on the board).
Take the box cover off and use the diode test function of your meter to test four of the five diodes. (Diodes have black bodies with a white stripe on one end, the "negative end.) The diode you can't easily test is the one closest to the black input wire because it has a 15 ohm resistor in parallel with it. The meter should show around .55 volts when the black probe is on the diodes negative metal leg, and the red probe is on the diodes positive metal leg. The diode with the most stress on it is the one closest to the red output wire. Some of the more expensive digital multimeters have a capacitor test function. You would need one that can read up to 1uf (1 microfarad). You can't fully test the SCR with a multimeter. The diodes can be replaced with diode 1N5399, and the SCR is a C106D, and the capacitor is mylar or polyester 1uf/250v. Mouser Electronics (online) has all of them.
Radio Shack sells a basic soldering iron with some solder for only $11.