Can someone spy on my IP address?

What can someone see with your IP address?

The website you were using may not have used SSL encryption (https://) and the information you send back and forth with them may be in plain text (unencrypted). If it was you probably can read it in your web browser.

If it's someone else, they'd need to be viewing a page on your domain that has all sorts of goodies including JavaScript. That includes some information sent back and forth between you and that site to establish a session cookie, so the answer is probably "a lot".