Bike broke down at shops

What goes to the ignition switch? The connector thing? But that's in the other direction.
The small red wire seems more likely to be the wire from the ignition switch but if that's the case it doesn't make sense to me that none of the lights or electrics work without that connected as well.

 

The red wire with the round tag goes to ignition switch. That is why if it's disconnected from the solenoid everything is dead as it's taking 12v from the thick red wire.
Everything operates through the ignition switch so no power to switch means no power to anything.
The thick red wire is coming from the battery and the thin red wire has to be attached to it so I presume it used the same bolted terminal as the thick red wire on the solenoid.
Just to recap the thick red wire is coming from the battery,it then goes through the solenoid to the thick black wire which goes to the starter. The thick black wire is not a battery earth wire.
If you wanted to bypass the solenoid because of the problems you were having you could disconnect the thick black wire(tie it outof the way) or just unplug the small connector.
The solenoid has to be operated by the smaller wires. Can't quite see in the pic but if those 2 small wires with the connector are attached to the solenoid then they are definately going/coming from the starter button via the main loom.
Basically so that you don't need a heavy cable going up to the starter button the smaller wires are used. These operate the solenoid which is just an electrical switch. The heavier current then flows through/across the contacts to operate the starter. This is what you can sometimes hear clicking if it's faulty or the battery is low.
I would guess from your original symptoms the solenoid needs replacing because it is sticking in the on position or it's getting power from somewhere either a short or faulty button.

 

ok first to understand what a solenoid is and doing.

inside the soleniod is a post with a t-bar ontop , around this post is a coil of copper wire. above this are two flat tabs with an airgap between them one is a pos and the other a neg they lead from your battery to the starter they tend to have huge thick chunky cables. you will also find a little connector block with two thin wires, these thin wires travel to a magnetic switch ( the post copper coil) in essence a electro magnet.

when we press the starter button we charge up the coil on the post, this crates a magentic field and the post shoots upwards allowing the airgap between the two main contact points to be bridged , this bridging closes the circuit to the starter motor and it starts to turn, when you let go of the starter button the magentic field generated ceases and a small spring overcomes the force and parts the T-bar contacts stopping the starter motor turning over.

small connector block is the circuit the starter switch opperates to open and close your solenoid.
red cable with eyelet looks to be a battery connection.

when a solenoid plays up its not possible to fix them, they are a sealed unit so its a throw away replaceable item.
click but no starter motor run, that click is the electro magnet shooting the t-bar upwards but over time they get worn out, from arc spark when the two contacts meet. or wear from the constant coliding of the t-bar against the two contact pads
Click and a buzzing sound , the click again is the electro magnet doing its thing but the buzzing is the contacts not fully bridging , you may find intermitent starter motor working then not working. in these cases the solenoid should be replaced.

the solenoid usually wears before the starter motor itself . if the stater motor fails its usually down to the carbon contact brushes against the comunicator on the center pole of the motor shaft. usuallly you may find a carbon brush kit to fix the starter motor and at the same time you would either lathe or rub the comunicator to clear it of any carbon build up.

 

Completed continuity tests on positive and negative cables from battery to solenoid and there appeared to be no problem. Any other ideas?

How do I check groundings?

 

There are NO positive/negative cables to the solenoid from the battery.
The thick red wire is the positive feed to the solenoid direct from the battery. When the solenoid operates the red wire connects to the black wire which takes the 12v positive down to the starter motor. The starter motor gets the negative from the earth on the engine.

From what you have said the starter is sticking in?
It can only do this in one of two ways. If the solenoid is stuck in it will turn the starter motor even with the ignition off. The other way is via the starter button if that is stuck or you have a short BUT when you turn the ignition off this would disconnect the solenoid.
My bet is the solenoid has burnt contacts or it's just not returning under the pressure of the internal spring.

With both thick wires connected to the solenoid if you put your tester or bulb + on the thick black connection on the solenoid and put the - negative on the engine you should NOT get a reading. If you don't get a reading that's fine as the solenoid isn't stuck in.

If you connect the thick red/black wires to the solenoid and the starter tries to work you have a faulty solenoid.

If you don't get any of those 2 results you can try and bypass the solenoid by taking the thick red /black wires and touch them together Firmly BUT only for a moment. You will get a spark but the starter should try to turn. ( no need to turn on the ignition) and you can leave the thin red wire off.

 

Notice how the battery is connected directly to the solenoid. The black "T" shape is the moving contact. When you push the starter button the solenoid coil engages that contact across the 2 terminals and the 12v + positive then travels down to the starter.
You can see from the diagram that 12v is always connected to the solenoid but doesn't go anywhere until the starter button is pressed.
You can also see the spring wrapped around the shaft of the "T" .

On the diagram you can also see the ignition switch with a wire going to the battery + that is your thin red wire with the tag on the end which can either go to the battery + (as in the diagram) or the thick red wire on the solenoid connection which may be your way??