I got some new wiring harness & it didn't come with the connector to this piece.

Thanks 

That my friend looks like your turn signal relay.

^

Thanks so that rules out one thing for my bike not starting 

I see taped wires on the side stand relay (to the right of the turn relay).
What are the symptioms? Nothing happens? Lights come on?

There has to be a 2 prong connector for the turn signal/flasher relay. The relay to the right of that is  the side stand relay and required a 4 pin connector. If side stand relay not energized, the bike is dead. That relay is controlled by the side stand and neutral switch along with the dual diode. With bike on center stand, put it in gear. As you move the side stand up and down you will hear that relay clicking. Relay is ON if side stand up or bike in neutral. In gear and side stand down, relay goes off and bike won't run or start (though lights will come on).

Right now the turn signal lights don't come on. I'm focusing on starting up my bike & I thought the turn signal relay had something to do with it (before I knew what it was).

Bike cranks but won't start. It sounds like it want to start really bad but one little thing is stopping it. 

Ok. You have lights and you have the starter motor going.

Do you have spark?

Side stand relay in your photo doesn't look correct - there should be a black 4 pin connector there.
The +12 to the coils goes through the side stand relay. If side stand relay not connected right, you might not get spark.
That +12 then goes through the kill switch on its way to the coils.
You should not have the start button working of the above is a problem because the start button gets its +12 to run the starter relay from the same circuit.

You should see +12 on one terminal of the coils.

I think you need to get out tour wiring diagram, volt meter, and start tracing connections. There aren't that many wires in a GS500. All wiring is pretty simple. The Haynes wiring diagram is in color which makes it super easy to follow.

Hey Adidasguy thanks for the tips. I'll check my side relay.

Let me give you the story of what happened.

About 2 months ago after work my bike wouldn't start. I had to push it home on Christmas. So I was thinking it might be wiring or at least if I change my harness then I can put on the stock coils (bike had aftermarkets). Bike did crank before new harness just wouldn't start. Same problem I'm having now. Yesterday I clean the carb real good. A few more things I gotta look into. Checking the fuse & change the fuel lines.

Most important now to check for spark.

No spark could be:
1. Bad connections to coils. One side of coil would be +12. Other terminal goes to the ignition control.
2. Bad ignition timing pickup
3. Bad connection from timing pickups. Follow that wire!
4. Defective ignition control unit or the small ground wire from the battery is off as that is the ground for the ICU.

#2 is least likely as it is rare for both pickups to go bad. At least one cylinder should try to fire.

If you have spark from both, then we need to dig deeper.