How do I connect the wallet to a website?

The website asks, not you. When it does, the wallet opens its own window naming the site that is asking and the account that would be used, and you choose to agree or refuse. Nothing is shared until you agree, and refusing is reported back to the site as a rejection rather than an error.

Connecting lets a site see your public address and ask you for things. It does not let it move anything: every transaction and every signature is a separate request with its own confirmation screen that you have to approve. A website cannot spend from your wallet on its own.

Every confirmation names the requesting site at the top. If that name is not the site you think you are on, refuse.