Removing the lamps (bulbs) from each fixture and then testing one-by-one could narrow it down to one fixture or, if it blows the circuit no matter where the bulb is, to the switch.

Circuit breakers rarely go bad.

Also, it's kind of hard to tell without knowing your switching diagram: are all the lamps controlled by the same single switch?

Is it by any chance a three-way switch (controllable from two different wall switches)?

Note to Conservative Cavers: Yes, I'm also an electrician, ya assholes.

Thanks!

the lights over the vanity are all controlled by a single switch - 4 lights on above each vanity.

This sort of style, though ours is silvery and without the pine cone motif.

I agree with NYC SKP for the most part. Breakers do go bad in very damp or very dusty environments.
But it's not the breaker.
You may have shorted the lamp socket when you removed the old bulb. It happens.
It's not the switch either.
Take out the new bulb and turn on the light. If it lights, it's a shitty bulb made in China. Try another. If another bulb trips the breaker, it may be the fixture.
First:
Make sure the vanity light switch is off. Then:
Open the fixture that has the bad bulb.
Disconnect the black wire coming out of the wall and cap the end off with the wire nut.
Turn on the switch.
If the other vanity light goes on and the breaker holds, you have found your problem.

and the first reset of the circuit breaker, I changed the bulb on the offending light above the vanity and left it out and when I flipped the vanity light switch, the whole room went dark again... I then tried putting a new bulb back into the vanity light and retried it again, and the vanity light switch again caused the whole room to go dark again. (so, I've been up & down the stairs a lot today...)

Unfortunately, when I tried taking the bulb out of the vanity light this time, I think I also broke the vanity light.

Follow the directions in my previous post and disconnect the offending fixture. I'm assuming it worked fine before you changed the bulb. If the other vanity light works, you have found the culprit.
If you haven't touched the other vanity light it is not likely the problem.
Bathroom vanity light are very pretty and useful, but they aren't very durable and are made from very cheap parts. You likely damaged the socket when you removed the old bulb and caused a short in the fixture.
I really doubt the problem is anywhere else.
Just make sure when you remove the fixture that you cap off the black and the white wires coming out of the wall so nobody gets hurt.
This is the best advice I can give with the limited information I have to work with.

If you like, you can PM me and I will be glad to help, okay?

Part of the problem is that there are two vanity lights, one over one sink on the left wall of the bathroom, another over the other sink on the right wall of the bathroom. (see my picture above - two light sets like that). All of those lights went out, as well as the main light. When I flipped the breaker, the main light went back on both sets of vanity lights stayed out. I've tried the "offending" light socket with two different new bulbs in it as well as with no bulb, and all 8 vanity lights (4 bulbs on each side) don't work. It just seemed odd to me that the main light is on, but both sets of vanity lights stay out.

I guess that is progress.

But I think CallMeCrazy is right. The problem was in that fixture.

Sometimes I can't see the forest for the trees.
It's not the breaker and very unlikely the switch. The breaker is doing its job. The switch replacement would require someone who is comfortable working with electricity to replace it.

But it's the fixture.

Last summer, I had a really peculiar electrical problem. I had a circuit that had only a dehumidifier plugged in. It started working intermittently. Because the breaker never tripped, I figured that had to be a problem in the dehumidifier. But I moved that to another circuit and it worked perfectly.

I put other things on the bad circuit and they worked fine. What is going on here?

The dehumidifier might work for 10 minutes and then the circuit would be dead.

I figured the CB must be going bad even though it never tripped. My wife will let me work in the gang boxes, but she won't let me open up the main CB box because she's afraid I'll burn the house down. So I called out my electrician. It ended up being a loose wire going into the CB. As long as it was cold, it completed the circuit, but as it warmed up, it moved just enough to open the circuit.

Good thing because that was an expensive CB -- not the kind you can buy at the hardware store.

Electrical wires vibrate as current passes over them. Over time, they can work themselves loose. The process usually takes years but it does happen. The first thing I do if I have to open a panelboard is tighten all the screws on the breakers and the neutral bus. That way I know the problem not because of a loose yoke.

I had an electrician out a couple of years earlier. If he would have tightened all the wires I could have avoided that dehumidifier problem.

And now I need a plumber to explain how a toilet that hasn't been touched for over a month suddenly decided to turn the supply line into a fountain and dumped 40 gallons of water that ended up going through the floor into the basement at 3AM.

I believe the canalization below the house or in the garden was clogged and the rain water from the roof had no way to go but back into the house.

a friend of mine that is pretty handy also mentioned resetting the GFI switch in the bathroom to see if that works. He said he's gone chasing electrical problems a few times in the past and all he needed to do was hit the reset button...

So, I tried that and it worked (one light on the vanity lights is not working.)

You may have a dangerous situation there.