Turning key is a crap shoot
ground the starter relay to the body. metal on metal
Just make sure you get the right terminal. Some cars, those two "push on" connectors, can be reversed.
Take a test light, see if you can identify which "push on" relay wire is YELLOW this is coming out of the bulkhead from the ign switch. Put your light on that wire while hooked up and it should light when key is twisted to start.
As 73 said, ground the other terminal. On automatics, this goes down the firewall to the transmission neutral safety switch, on stick cars it goes into the interior to the clutch switch
It could easily be a bad relay, a bad terminal in the bulkhead connector, or a problem at the clutch / NSS