The easiest day trip from Tórshavn has to be the village of Kirkjubøur. One of the free public buses from Tórshavn goes there. More info about free buses here. The journey takes up to 20 minutes and you can explore this beautiful historical village and enjoy the views over neighbouring islands. Just watch out for the strong wind. It was super windy when I was there but still worth it.
Kirkjubøur is probably the most historical site in the Faroe Islands. It is the home to St. Olav’s Church, the oldest still used church on the Islands, dating back to 12th century. A runestone has been found inside St. Olav’s church in 1832. The Kirkjubøur stone is now in the National Museum of the Faroe Islands in Tórshavn. Another historical site is the Magnus Cathedral. The historical home to the Roman Catholic Diocese of the Faroe Islands until the Protestant Reformation in 1537. It is now ruined but it was built around year 1300. There is also Kirkjubøargarður (Kirkjubøur yard) which has always been one of the biggest farms in the Faroe Islands. The oldest part of the farm is the oldest wooden house in the world and roykstova (front room without a board ceiling, where the hearth was situated and with a smoke hole in the roof ridge) which dates back to 11th century. The farm is still inhabited by the same family since 16th century.