One of your first stops should be Ekenas Visitor Centre. Here, you'll be able to find your bearings and get lots of useful information before you begin exploring.
You'll soon find out why this Ekenaes tourist information center is a hit with travelers. If you're looking for other things to keep you busy, Ekenas Museum Center and Church are in this area as well. They won't take you very long to reach from Ekenas Visitor Centre.