What’s the process of getting a car loan from a bank?

So when I have enough saved up to pay off my current car loan, my wife and I want to get a brand new car.

Dealer loans tend to have pretty bad interest compared to banks like JA.

But how does getting a loan from a bank to buy a car work? With a dealer it’s easy.

Do I go to the bank first? Or the dealer to get a quote?

I don’t want to waste the dealer’s time.

If I go to the bank first, do I get a range of what I’d be approved for and use that to purchase a new car?
What about trade in for the current car? If there is any balance remaining on the current car’s loan would I need to pay that off or could I use some of the balance of the new loan to cover what I owe?