I can safely say that Pythagorio is one of my favourite spots in whole of Greece. The town is about 1500 people strong and has a perfect mix of tourist infrastructure, beauty and authenticity. It has a stunning harbour which is occupied with boats of all sizes. During peak season you may also see one or more multi-millon Euro yachts here, which leads to a bit of Marbella feeling (in the good way). The whole marina is surrounded by bars and restaurants mixed with a few souvenir shops. I have to admit that we have never eaten in this area although the tavernas are pretty crowded. I think the main reason is that i personally really see this as the bar area where you can waste an envening watching people walking by and spying a little bit on the rich and famous on their massive 50m 4 story boats.
Pythagorio does not have much of a beach to be honest. There is a small beach at the end of town which is great for the food the tervernas serve (e.g. Faros), but probably only 5 meters wide. I would recommend to go here for dinner as the staff will remove all sunbeds and put up tables right at the beach. For daytime swimming you will need to look outside of Pythagorio. The next beach worth mentioning is Potokaki. This is actually the area where most of the hotels around Pythagorion are located. It is right next to the airport and you can see (and hear) planes landing if you fancy that. The area also has some popular new beach bars which are attracting lots of young Greeks.
What makes the village such a favourite of mine is actually the charme it develops at night. There are dozens of very nice tavernas to choose from of many are located in nicely equipped gardens. You will surely have enough choice here for a two week holiday and typically the food is always nice. Samos cuisine is rich and the chefs make a lot of use of all the great herbs that grow on the island. You will find really nice shops that range across jewellery, clothes, pottery, herbs and tools. Actually the tools shop on the main road is worth a visit. This is a place which is packed with things right from the floor to the ceiling in a wierd and almost artisitic way. I am pretty sure you will find almost anything in here from a hammer and nails down to spare parts for motorbikes and electronics equipment. The owner will probably produce whatever you require from some deep corner of the shop or order it for you if it really should not be in stock. The main road actually gets closed for traffic after 6p.m. and this makes it especially nice to stroll along the shops.
I prefer to stay inside of Pythagorio and i can actually recommend a few places which are family run, just search this blog for more information.
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From Pythagorio you also has access two of the best beaches on Samos, Psili Amos is just a short boat trip away. The real gem though is Samiopoula, a small island about 1h away from Pythagrio with a lovely sandy beach.