Ignition lights turn on, engine won't crank. How to perform trouble shooting?

Question

I have a 2001 Lexus RX300. Ignition seems to work, but the engine just doesn't crank. I'd like to fix this problem myself.

Here's some details:

1. When the key is turned to "on", the lights come on. When the key is turned to "ignition", the lights all go black, and sometimes there is a very quiet click in the engine compartment, sometimes there is no click. When the key is turned back to "on" from "ignition", the lights remain black. The key needs to be removed from the ignition for ~30 seconds for this to happen again; otherwise, it just stays black.
2. The battery voltage remains at about 12 V throughout the entire process.

What is the process for diagnosing a problem? There are a lot of suggestions on the internet about what it could be (loose ground cable, bad starter, bad coil, bad ignition switch) but I'd like to narrow it down a bit.

Please help!

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Edit 1

Another detail:

3. When the key is "on", and I push the brake pedal, the lights immediately shut off. When the key is "off", and I hold down the brake pedal, nothing happens.

I attempted to reset the immobilizer at the suggestion of @MTA. Here are the steps:

a. Get out and close driver's door. Insert key in driver's door lock.
b. Turn key to lock.
c. Turn key to unlock. Remove key.
d. Press LOCK on remote.
e. Press UNLOCK on remote.
f. Get in the car and see if it starts.

Before I could do this, I need to mention that the "Master Power Window Switch" has been broken for more than a year. I had to reconnect it in order to use the remote. I did this, and the remote seemed to work.

Answer 1

Now that you've added the info in EDIT 1 (dash lights go out when you step on the brake), it's likely that you have a poor, high-resistance connection at the battery. The resistance between the battery terminal and the battery cable is so high that current can only trickle through. When you try to increase the current by activating the brake lights, the voltage drops to an unusable level.

If the connections at the battery terminals are actually clean and tight, then the culprit would be at the other end of the thick cable that connects to the (-) terminal. That other end might be bolted to the engine block and a thinner wire from the same (-) terminal may go to another ground stud in the engine compartment.

Remedy

For each battery terminal, you should unclamp the cable, clean the terminal and the inside of the clamp with a wire brush till they are shiny, then re-connect the cable, tightening the nut well. All wires to the battery terminals must be clean and tightly connected. Nothing must move if you tug on it by hand.

Suggest you do this before further diagnosis because loss of dash lights from stepping on the brake is either a bad battery connection or a nearly dead battery. Since you've measured 12V at the battery, that leaves a poor connection as the cause.

Answer 2

Before you go down the rabbit hole of diagnosis, let's see if the immobilizer has lost its mind. You can reset it yourself easily. You need your key and your remote.

1. Get out and close driver's door. Insert key in driver's door lock.
2. Turn key to lock.
3. Turn key to unlock. Remove key.
4. Press LOCK on remote.
5. Press UNLOCK on remote.
6. Get in the car and see if it starts.

If it doesn't start, let us know and someone can make another answer to begin diagnosis.