I have recently set up GPG to sign my Git commits so now I have a signingKey field in my gitconfig. I'm not very familiar with details of GPG – is this signingKey a sensitive piece of information that I should keep private or does it fall into the public part of gpg? I have my gitconfig in a public repo where I keep my dotfiles and I was wondering if it's ok to have that field visible.
No, it isn't necessary to keep it private.
The secret key is not in git's configs but in the GnuPG's "keyring", which is usually some file in your HOME. In theory it can also be in more secure locations, like hardware token, but I don't know much about it.
The value in git config only instructs gpg which secret key to select.