Starter troubleshooting
There are only a few ways that a starter can fail:
1) The starter or its solenoid are bad and need to be replaced.
2) High-amperage 12v current can't make it through the fat wire from the battery positive terminal to the starter
3) The ground connection is bad so the high-amperage 12v can't make it back to the battery again.
4) When you turn the ignition switch, it's failing to connect 12v to the solenoid (When you have 12v at the solenoid it should pull the starter against the ring gear and connect the fat wire to the starter motor so that the current in #2 above can flow and spin the engine).
5) On US-spec cars, the seatbelt interlock circuitry (stops the car from starting if you haven't fastened your seatbelt) is is screwed up any of many different ways. This is actually a special case of #4, but is treated separately because the cure is to bypass the interlock circuit instead of trying to figure out what's wrong with it.
Robinsonx1/9's problem is definitely not #4 because he's hearing the solenoid click. It might be #1.
DjPatD's problem almost certainly is #4, just as popsprocket says, because he's not getting a solenoid click. It's very unlikely to be #1 because he's already replaced the starter.
But if you want to deal with this in a systematic way...
First, stupid stuff: Are the bolts holding the starter to the bellhousing tight? Are all of the electrical connections at the starter, including especially the red wire with a slip-on spade connector, clean and tight? Does the battery show 12 or more volts between its terminals?
Second, check the ground strap from engine block to chassis and the ground wire from battery negative to chassis. Both should be solidly fastened at both ends and reasonably free of corrosion. Use an ohmmeter or a test light to check that you have decent continuity from engine block to the negative terminal of the battery.
Third, check the voltage between the starter terminal where the fat wire from the battery is attached and the engine block. It should be near as no never mind what you're seeing at the battery.
Fourth, remove the spade connector from the starter and see whether that wire is delivering +12v relative to the engine block when the key is turned.
Fifth, jump the spade terminal on the starter directly to the positive terminal of the battery. This will make the motor crank if the starter/solenoid is good, the grounds are good, and the heavy connection from battery positive to starter is good.
Check all of these and you'll find what the problem is.
