from the problem you describe, it sounds like you have a bad switch.
so far, you are doing a great job diagnosing it. the next step would be to check the start button and on/off switch.
first, when you jump the wires on the relay, and all switches are set to "ON" and key is "ON" does it start or only crank the engine over without starting?
if the bike does start when you bypass the relay, the start button is most likely bad. if it does not start, the problem probably is your on/off switch
to test the start button, you will need to be able to access the wires for the button. you can either open the start button switch, or follow the wires to where they go to a connector. to test it with a multimeter, unconnect the switch, and put the multimeter leads into the wires THAT GO TO THE SWITCH. if you put the multimeter leads into the harness instead of the switch you will get a bad reading. place the meter on OHMS(looks like a horseshoe)setting and 10 range if not autoranging. with the button pressed and the meter leads in the wires, you should get 0 ohms. with the button not pressed, you should get OFL(OFFLINE), or no connection. if you get these results the switch is good, and plug it back in. if you dont have a multimeter, with key on and on/off switch on, you can jump the wires for the start switch. if the bike cranks over now, then you know the start button is bad.
to test the on/off switch, it will be the same testing procedure. the easiest way to test it is to set the multimeter on DC volts in the 20 range. place one lead on the wire that goes from the on/off switch to the start button. place the other lead on a good ground, like the battery terminal. if you have the key on and on/off switch on, you should get at least 10 volts on this wire. with the key on and switch off, you should get 0 volts to this wire.
test these switches, and i believe you will find out which is the bad component.
if both switches are working fine, then your start relay is bad
i see you got a new one from radioshack, but does it have enough voltage to work properly?
an automotive relay might work better.
did you test the switches?
when you push the start button, as normal, do you hear a "CLICK" noise? or nothing?
have you checked you get battery voltage through the start button and on/off switch
okay, that helps
the red/yellow wire should have battery voltage espically when you are pressing the start button. if you are not, then the wire for that particular circuit is bad, which comes from the key switch.
so if you dont get power to the start button when holding it down, you must replace the key switch. that also explains that after you replaced the start solenoid, nothing changed.
you can verify this by exposing a small spot of the start button circuit, and applying battery voltage to it when the key switch is on. that will give the circuit voltage where it is supposed to be, so when you give it battery voltage(like it should have) and press the start button the bike will start.
the cdi ignition, or ECU, should have at least one power source and one ground source.
did you test the start button by applying battery voltage to the wire to see if it cranks over?
because you bike starts and runs, the cdi is not the problem.
the bike has numerous different electrical circuits: starting circuit, charging, ignition, lighting, horn, signals, etc.
the main purpose of the ECU is to read all the signals from the sensors, like crankshaft position, throttle position, and engine temp, exhaust temp, etc and then the ecu decides when to make spark happen and when you change the ignition curve. sometimes these circuits will share some wires, like power or ground sources, but never share signal wires. each component will have a dedicated signal wire, for example the crankshaft sensor sends an AC voltage current to the ECU. by the speed and time of the signal, the ECU can determine when to spark the plug and how fast.
test the start button how i have described, and if that works then your key switch is bad, and needs replacement.
are you sure you hooked up the replacement relay correctly?
hopefully, my information is helpful. i am trying to describe everything as best to my ability.
good luck, thanks