A great way to appreciate the true Skiathos is to explore on foot - particularly in the hills. About 95% of tourists, I reckon, spend their entire holiday on the beach, by the pool - in other words, never venturing anywhere beyond the south coast.
This is fine, but if you need a bit of head-space, some peace and quiet and some fantastic views, head up to the hills! There are two roads leading into the hills - one takes you up before the Kalamaki peninsula if you're heading from Skiathos town, and the other, larger road, turns off just before Troulos (this is the one we did).
If you do explore on foot, ask the locals (taxi drivers are a good bet) where the best places to go to are - there should even be some minor expeditions into the hills, try asking your holiday company.
The remoter parts of Skiathos are gorgeous, miles of pine forests and a number of monasteries - plus fantastic views. And, there's no people! Be careful of the sun though, and take plenty of water.
The north of Skiathos is rarely explored by tourists, apart from the famous Lalaria beach, accessible only by boat. We took a taxi to the Panayias Kounistras monastery, and walked the dirt track through the hills to Mikros Aselinos beach, a quieter version of Megalos Aselinos.
The beach is beautiful - when we arrived at lunchtime there were about ten other people there, by 4pm there were only two other people. There's also a taverna selling soft drinks, ice cream etc.
It's worth putting some extra effort to explore the north of the island; we were rewarded with a near-deserted beach. The contrast with the ridiculously over-crowded Koukounaries couldn't be greater.
The beaches around Koukounaries are definitely the best of the entire island. Superb beaches with unbelievable turquise water, lush vegetation, but sometimes crowdy. I really liked Aghia Eleni beach. This beach is a mix of the organized beaches like Aghia Paraskevi or Troulos and wild northern coast. The atmosphere was really chilled out, sophisticated or pleasant, hardly to select the most appropriate word. You can watch the stunning sunsets, long legged Scandinavian girls or sit in the beach taverna and listen Jaques Brel or Charles Aznavour songs. It is a place for young couples.
Similar feelings, but more lonely, are to be awoken in the Mandraki beach. The sea could be rough there, but the beach is very attractive. The further to the north you are walking, the more the scenery becomes wilder and less civilized.
Western shore is completely another Skiathos as it is known from tourist brochures - remote, dramatic and particular rocky.