Went to Fourni last summer from Ikaria and stayed 5 days. There is a boat that goes to Fourni every morning from Agios Kirikos, about 1hour trip. Our room was close to the port, so we had everything in a walking distance.
The second day we took the tour boat that goes round Fourni and stopped in Chrissomilia, a nice beach with pebbles but clean water. There is a tavern there with great recipes. Unfortunately, the rest of the days it was too windy and the boat didn't make any journeys. But one day we walked about 20 min to Psili Ammos and the rest days we just laid our tired bodies in the beach close to the village. This is a good island for naturism, hiking and lots of relax.
Not many people go to Fourni for holidays, which means too quietness and cheap prices. Try all the taverns at the port, they are excellent and the people seem so hospitable. I loved the late evening hours, about 10 to 12 at night, when people had finished dinner and they were walking up and down the port. You could hear cheerings and hellos, accompanied with hugs and handshakes. Really great place and I have nice memories from Fourni.