Rome is amazing, one of my favourite cities. I grew up with a history-mad mother, with a particular interest in the Romans, so I've been brought up to seek out Roman sites here, there and everywhere. My DD has also now grown into a fascination with the Ancient world, so particularly over the last few years, we've ensured we get to see important sites when we travel (she now studies Archaeology and the Ancient World at University, so it's clearly her 'thing'!).
Athens is also amazing, and we travelled around Greece, seeing Olympia, Delphi, Mycenae a few years ago (I'd done similar on a school trip about 250 years ago).
We visited Turkey last summer, mainly to see Ephesus which is very impressive, but Istanbul and Turkey in general left me slightly cold I'm afraid.
There are a wealth of ancient sites across Europe, we're very lucky, I've yet to get to Herculaneum, but Pompeii blew me away. But then there are so many fascinating places dotted everywhere - Tarragona in Spain, Nimes/Arles in France, Bath in the UK, wonderful floor mosaics in the museum at Cirencester. I could go on and on.
However, beyond the ancient world, there are so so many wonderful cities in Europe: Amsterdam, Bruges, Lisbon, Barcelona, Budapest, Salzburg, Vienna, Munich, Berlin, Paris, Prague, Florence, Bologna....I've done a bit of interrailing which is a really easy and quick way to see lots of cities over a holiday.