Having grown up listening to stories of Greek Mythology such as Hercules, Perseus, Theseus... a lot of 'euses'... I have always held a love and deep fascination for anything to do with Greece, too. That would be an amazing place to visit, and I would dearly love to see the Temple of Athena up close.
My favourite ancient civilization to read about has always been ancient Egypt. I remember learning about it in school, and I found it super interesting. I love hearing about the Pharaohs, their complex tombs and burial rituals, their mythology and their ancient artwork.
I don't know if I'd necessarily want to visit Egypt, but it might be interesting to see the few things that were left behind by the ancient Egyptians.