Starter or Ignition Switch Problem?

I have a 2000 Toyota Avalon with 185000 miles. I went to start it this morning. It acted like it was going to start easily as usual.
Then it acted like the electrical system went out. There were a few clicks when trying to start and then nothing.
No electrical power at all. In what I guess was less than 30 seconds the electric power came back and I was able to start the car.
The battery is new.
I know the power was totally interrupted because I lost my presets on the radio station and the dash lights were off.
I am thinking that something shorted out and then it reset. Can that happen?
I very gingerly use the ignition switch now when starting the car. I turn it on, then only twist very gently.
I am a little afraid that it is going to die on me again.
Any suggestions?
Thanks.

I know you said the battery is new but that is where i would start looking. i’ve seen new batterys fail it sounds like an internel short circuit inside the battery. then check the starter while your at it check the cables for a loose contact it might be something small.

I very gingerly use the ignition switch now when starting the car. I turn it on, then only twist very gently.

I doubt it is your ignition switch. If that didn’t work, it wouldn’t cause your car to lose its presets. It also didn’t short - if it did, you’d see huge sparks flying everywhere. You have an intermittent open or high resistant connection somewhere.
Check the battery cables, as Marc says. Be sure to check the other ends as well.
If the battery is old, it could be the cause as well but if it is new, that’s a bit less likely unless it has some sort of defect but first check all your connections.

I can’t thank you guys enough.
WalMart had just installed the battery a week or so ago.
The positive connector was loose as a goose. It was easily swiveled around and I could have simply lifted it off of the pole.
It appeared that it probably was not pushed down onto the terminal prior to tightening. I loosened the connector and shoved it down and retightened. I think I am good to go now.
BTW this is the 3rd Walmart battery. I have had 2 go out in less than 3 years, so I get the whole purchase price back. The bad news is that have generally gone up about $25 in the intervening period, so I have to pay that.
Thanks again.