I have my favorite spots on the island and am on a constant lookout for new ones. Word of mouth is, as always, the best, but if that fails, I refer to a couple of publications.
"Short Nature Walks" by Rodney & Priscilla Albright published by The Globe Pequot Press, contains inviting, short and sweet descriptions of various towns, former estates, parks and preserves, beaches, marshes and bogs, whatever they are.
"Hiking Long Island" by Lee McAllister published by New York - New Jersey Trail Conference, lists established trails. This is a serious hiking guide, taking great care to cover all important information. Admittedly I have only attempted shorter hikes listed there, but they were not disappointing.
If the weather is an issue you may want to pick up "Where to Go & What to Do on Long Island" guide, which lists many historical places, museums and other attractions, greater and smaller, a lot of them indoor. This book is out of print.
Obviously, you can get these and other books from Amazon, but why not try a local bookstore. There are still some around. Since we are on a subject of favorites, my favorite is Book Review in Huntington, a very active bookstore with many famous authors holding book signings there.