I might be a little biased about Athens because it's where I went on my first time ever in Europe, but I can't say enough amazing things about this place. Whether or not you're a history buff (I'm not), the history surrounding this city (and country) is mind-blowing.
The Acropolis. The National Archaeological Museum. The Olympic Stadium. Whatever your motivation behind visiting Athens, please go check out some of the ruins. Any of them. All of them. Note: If you're making a trip to the Acropolis, I recommend wearing shoes or sandals with good grip. Flip flops or ballet flats might not cut it - the marble steps and walkways can be extremely slippery. I couldn't help but take a moment, sitting on the steps of the Acropolis, to just put my camera down and really take in what I was looking at. Thinking about how Athens was built and the work that went into creating the city, all I could say was "Wow".
While much of the culture of Athens is built on history, what is culture without food? Feta cheese, souvlaki, tzatziki... yum! One of the best meals I've had in my life came from Sissifos Tavern in the heart of Plaka. This restaurant has both an indoor seating area and an outdoor rooftop patio, with a remarkable view of the Plaka area and the Acropolis. Service was excellent, and the food was beyond delicious. Plus, if you're not exhausted from spending the day exploring, the Plaka area has many shops and bars to wander through.