Rye Neck schools are excellent!, their schools are more like private schools than public. You will use the Hommocks Pool which is an indoor/outdoor poll in the Mamaroneck School District, Mamaroneck Harbor has a beach available to all residents of the village, there are also a few private beaches used by communities on the waterfront in Mamaroneck.
If you live in the Rye Neck section of Mamaroneck it means you are a Rye town resident and can use the beach in Rye (i.e you pay the same beach fees as Rye City and Rye Brook residents which is minimal compared to fees you will pay if you lived in Larchmont or the part of Mamaroneck in the Mamaroneck School district or live elsewhere in Westchester). , visit
www.soundshorerunners.com or .org for info on running jogging the area. The YMCA most people in the area go to is the one in Rye.
Most community activities take place at the harbor, with parades etc taking place on Mamaroneck Avenue. I am not sure if there is a center per se other than these but you can call the village to find out. The village usually has recreational activities year round including a summer camp.