My husband and I spent 3 nights on Ikaria about 10 years ago, in mid-September, as part of a 3-week island hopping trip, and we thoroughly enjoyed it. As others have said, it's not a typical Greek island, full of tourists. Instead it's very rugged, with mountains and lakes and sheer cliffs. Absolutely gorgeous. Very quiet. Very few tourists.
Ikaria has two ferry ports: Evdilos and Agios Kirykos (the capital). We arrived in Evdilos on a ferry from Samos and stayed in Armenistis for 3 nights. We rented a car one day to explore the island and decided that Kampos might have been a better base. It has a 12th century church and, nearby, the remains of a Byzantine castle and chapel that you can hike to.
Driving on Ikaria is easy. The roads are uncrowded and fairly well marked. You don't need to reserve a car in advance. Your hotel owner can help you set it up; ours was delivered to our hotel and picked up there afterward.
We had planned to ferry from Evdilos to Patmos, our next stop. But a mechanical problem delayed that ferry for a few days, so instead we took a bus to the ferry at Agios Kirykos, then back to Samos overnight, and on to Patmos the next morning. The bus ride to Agios Kirykos passed through some spectacular scenery along the coastline, reminiscent of the Amalfi Coast. We wished we had driven into this part of the island.
We didn't stop in Fourni, but if you're considering adding another stop, I wonder if you could fly into Ikaria and out of Samos for the same low price? This would allow you to take a day trip to spectacular Ephesus in Turkey ... a must-see, if you haven't already.