Walking around Athens can be pleasant and often the only way to get from one place to another in the historic center. An all time classic walking tour would start from the Acropolis Museum, which is an absolute must-see in the city. After appreciating the museum's treasures, cross Dionysiou Areopagitou street and you are in Plaka, the district of the gods as it has been named. Take Vironos street and browse in Krokos to shop for traditional Greek delicacies. 
Lysicrates monument — Photo courtesy of Nelly Paraskevopoulou
Continuing towards the center, you'll pass the famous Lysicrates monument and proceed on Tripodon street, which leads further into this picturesque district with its neoclassical houses, tavernas and souvenir shops. Turn right and downhill and you find yourself on Adrianou street, which leads to Monastiraki. Miltons, right on the corner, is ideal for a coffee or snack. The Roman Forum is on the street parallel to Adrianou, just below the Tower of the Winds, and is definitely worth a visit. Hadrians library is also on your way, right behind the old mosque on the central square.
At this point you are in Monastiraki and can walk down Hephaestou street, where all the curio shops are. Eventually the itinerary will lead you to the Ancient Agora, a major attraction in Athens and much appreciated by visitors. The majestic Acropolis can be seen from various angles during this walking tour. If you feel like a cool drink or lunch, Adrianou street offers many choices of cafes, snack bars and restaurants. By far the most elegant is Kuzina, opposite the temple of Hephaestus and the railway. Itinerant stalls sell everything you can imagine, from memorabilia to second hands books and handmade jewelry. All this area is pedestrian, so you can enjoy your walk without worrying about traffic. At the end of Adrianou street, turn right and cross Ermou street.
street art in a parking lot in Psyrri — Photo courtesy of Nelly Paraskevopoulou
On your left is the area of Keramikos with the ancient Greek cemetery, and straight ahead is the bohemian, arty neighborhood of Psyrri. 
Museum of Islamic Art — Photo courtesy of Benaki Museum
Visit the Museum of Islamic Art on Dipylou street, home to artistic treasures from the Islamic world. Explore the quaint side streets which seem to belong to another era and discover hidden treasures. Take your time and don’t worry if you get lost, this neighborhood is like a maze and eventually you will find your way out. Interesting shops are Sabater Hermanos on Agion Anargyron, which sells handmade soap in a great variety of colors and aromas, Komboloi tou Psyrri further on with handmade worry beads and decorative items in blown glass, Tonic Essentials on Pallados street, with cosmetics from all over the world and Karras leather goods on Miaouli street, where you can buy beautiful handcrafted bags and satchels. From the central square of Psyrri, take Aischylou street and taste the wonderful desserts at Serbetia. Lemon Poppy Seed is an avant-garde hairdresser on Protogenous in case you need a coiffure, and Paliosinithies on the same street sells second hand furniture and housewares at rock bottom prices. Excellent hotels in this district include Magna Grecia and A for Athens, with magnificent views.