Look a SURE way to chase down the starting problem is to go to the starter FIRST! There, connect a wire to the small push on connector and apply 12v to it. There SHOULD be at least an audible "click" of the solenoid engaging. If that does not happen then you have a bad solenoid or bad ground. Toyota does a good job of insulating the motor and tranny in rubber mounts and on the V6’s if there is not a good ground to the engine not even the solenoid will energize.
So, once you apply 12v to the solenoid directly what should happen is the starter should turn over the engine.
If you hear the solenoid energizing but no starter motoer at that point then most likely the contacts inside the solenoid. You can check this for sure by using a heavy guage wire at least 10g and touch the terminal lug opposite the battery lug, JUST the starter motor should spin without engaging the engine.
But since you say it is all new I would expect that applying 12v to the small connector will turn it right over but at least you can eliminate the starter.
The starter and solenoid will actually operate on pretty low voltage even below 10v.
If the starter works in this manner you really need to pick up the factory manuals for the correct wiring diagrams, NOT Haynes they are the worst when reprinting Toyota diagrams.
Now you can move on to your ignition problems.
Good Luck