Swedish postal service Posten has a network of around 3000 counter services in shops, petrol stations and supermarkets across the country. Look for the yellow post symbol on a pale-blue background, which indicates that postal facilities are offered.
If your postal requirements are more complicated (such as posting a heavy parcel), ask at the local tourist office. Package services are offered at certain office-supply stores.
Mailing letters or postcards weighing up to 20g within Sweden costs 7kr; it’s 21kr to elsewhere in Europe and beyond. Air mail will take a week to reach most parts of North America, perhaps a little longer to Australia and New Zealand.