The bash deprecation warning from macOS can safely be ignored, or you can add export BASH_SILENCE_DEPRECATION_WARNING=1 to ~/.bashrc` to permanently silence it.

The initial brew setup script you're using was deprecated, you'll want to use /bin/bash -c "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install.sh)". I've skimmed that script and I think it's actually zsh compatible too, but not 100% sure. This will set it up to be accessible by any shells, as long as you have /usr/local/bin in your PATH. (export PATH="/usr/local/bin:$PATH" in your ~/.zshrc, or path+=/usr/local/bin to use the zsh-specific syntax).

If you run which zsh you should still see some output; if your default shell did get changed some, you can change it back using chsh -s /bin/zsh.

EDIT:

I missed that you said you have an M1 Mac. According to the install script, the brew prefix is /opt/homebrew on ARM-based Macs (apparently this is to work around needing sudo for operations in /usr/local). I don't have a new Mac to test with, but adding path+=/opt/homebrew/bin to a new file at ~/.zshrc should to the trick.