I'm guessing that because your headlights are OK, your battery and its basic connections are probably OK, but as suggested, make sure they're clean before going any further.
And the fact that you hear the click of the starter solenoid means that its probably engaging, but you could have corrosion or other issues there, that's where I'd start looking first. Make up a heavy gauge jumper wire that is capable of carrying enough current to run the starter motor and jump from the battery positive terminal directly to the starter motor terminal (haven't done this on an ST so can't give you an exact location, but it should be obvious if you can get access to the starter motor). If you locate the starter solenoid the starter motor will be connected to one side or the other of the solenoid terminals. If the motor turns, then you can rule that out, if it doesn't, look for ground issues with the starter motor.
If motor turns, move the jumper to the side of the starter solenoid that is connected to the battery. turn on the ignition switch and press the starter button. If the starter motor turns going across the solenoid, you can rule that out next, and your problem is between the battery and the starter solenoid connection. If it doesn't turn (and you hear the solenoid click when you push the starter button) then you have solenoid issues, either corrosion or some sort of internal mechanical failure.
edit: because of the relatively high current involved, make sure your jumper is mechanically strong, with a strong metal clip at each end of the wire. Just pressing the wire against each connection with your fingers may not provide a good enough connection to flow sufficient current to make a valid test.