Buy a one-way ticket from Portland to Peaks Island, since the way back is free.
Biking is the best way to see the island, but you can also rent a golf cart (which is best if you want the shade). There are no public bathrooms since it's basically a small town; plan on making a food stop at a restaurant so you can use the facilities.
Since this is an outdoor activity, head out in the morning or around lunchtime to give yourself ample time to explore the island.
The island is beautiful in spring and summer, and even early fall.
Children under 5 ride the ferry for free.