What was actually filmed on Skiathos
The Old Port. The scene where Sam and Harry sprint along the dock trying to catch the ferry to Kalokairi — and miss it — was shot at Skiathos Old Port. It is unchanged and instantly recognisable, still lined with fishing boats and tour caiques.
The opening rooftops. Sophie sings 'I Have a Dream' while posting the three wedding invitations, against a backdrop of Skiathos Town's old rooftops. The belfry of the church of Agios Nikolaos and the clock tower are visible behind her. The viewpoint is up in the old town above the Old Port — the same climb that leads to the stepped lanes and the hillside restaurants.
What was filmed on Skopelos, not here
The wedding chapel — Agios Ioannis Kastri, near Glossa on Skopelos. It sits on top of a rock above the sea and takes roughly 200 steps to reach. This is the single most photographed location in the film and it is not on Skiathos.
Kastani beach — the main beach scenes. Also Skopelos.
Glysteri — Skopelos again.
The 'Dancing Queen' jetty — Damouchari, a village on the Pelion peninsula on the Greek mainland.
How to do it properly from Skiathos
Skopelos is a short ferry or hydrofoil hop from Skiathos Old Port — which means you depart from a filming location to reach the others. It works comfortably as a day trip, and most 'Mamma Mia tours' sold on Skiathos are exactly that.
Do the Skiathos scenes on foot before you sail: the Old Port dock takes five minutes, and the rooftop viewpoint is a short climb up into the old town. Both are free, neither needs a guide.