Beltsios Apartments (may be iffy spelling) in Troulos, on Skiathos. Absolutely gorgeous - we had a quiet apartment away from it all, but the pool bar was lively and fun without having music so loud you could hear it when you shut the door. It's family run, the people are lovely, can't do enough for you and it's a stone's throw from the beach.
There's only one bus that trundles up and down the island's only road, you can catch it into Skiathos town if you want nightlife, or on the other hand the Agustera bar in Koukounaris (more iffy spelling) at the other end of the island plays great rock music but unlike the touristy bits doesn't charge like a wounded rhino.
And as for running out there.......
It needs to be first thing in the morning before it gets hot, obviously, and it's hilly as feck, but the scenery's out of this world. The island's only road goes east to west, but you can run north on decent tracks and feel like you're the first person to discover the place. Well, until you find the taverna in the middle of nowhere!