Lycabettus Hill stands on the high streets surrounding the Kolonaki quarter. The hill area is wooded and has nice hiking trails. You can also visit the beautiful white church of Agios Nikolaos, St. George's Chapel. On the hill there is also an open theater and a restaurant with amazing views.
From here you can also just stand and look out over the entire city. The magnificent panoramic view extends all over the city to the sea. Take the time and try to locate the Acropolis and the Plaka. Try to locate the port of Piraeus and far from the sea, the nearby island Aegina
The stairs are not easy to climb, and in summer they are very exposed to the sun. Get a hat and water!