My personal opinion is to save the "fine dining" or "best restaurant" for places like Paris, London, or NYC. In Athens, I would recommend you a place that truly makes you feel like you are in Greece and are eating local food with the locals.
In Monastiraki Square, take one of the "shopping streets" that is NOT the flea market. Halfway down the street you will see bright lights and hear live music. Take a left and follow the short alley to a beautiful outdoor courtyard of about 3 restaurants, one of them with live music. The one I went to serve the best and most authentic Greek food at very reasonable prices. (It's not the money. It's the experience!) I don't remember the name but I have a picture of a napkin. The place was absolutely lively and vibrant and the vibes were friendly, welcoming, and chill. We felt so comfortable and we actually felt like we were Greeks for a moment.
