Getting to Moore Landing is either by car or by boat.
By car you can just drive over the Thousand Island Bridge (route I-81) and get off at exit 51 and head southwest a very short distance.
By boat, you need to go a bit farther away at the public landing docks in Wellesley Island State Park and then bike it east on bay Shore Road over to the Moore Landing area.
Alternatively, you may be able to locate some transient dockage at one of the nearby towns on the mainland (see http://www.nysgextension.org/marinas/stlmarinas.html) and then rent a car and come across the bridge.
Once you are in the Moore Landing area, you will find many people just enjoying a walk around the area or using a bike to go a bit farther afield.
If you are more venturous, you can rent a boat or canoe from the state park (800-456-2267) and paddle around the south end of the island.
You can always drive around the area too and many older folks find that just as relaxing: enjoying the scenery and waterscape from the comfort of your air-conditioned car.