Ardsley Schools offer music instrument instruction beginning in 3rd grade. Kid have a chance to learn to play strings or band instruments. I think the choices are sax, cello, flute, trombone, clarinet, drums, violin and some others that I can't think of. Lessons are once a week during school hours and are free though you have to rent/buy the instrument. If you are looking for private instruction, you can contact the Ardsley Music teachers for recommendations. The music instructor for the elementary school is Carol Ebenizer, and for the middle school, its Frank Squillante. Ardsley Schools main # is 914-693-6300.
For Swimming lessons, the middle school has a pool and they offer instruction ever Saturday. Click on the link that says Saturday Children's Instructional Swim
Ardsley UFSD > Rec. Pool Program
Greenburgh also offer instructional swim to non-residents through their recreational program. You can download a brochure here. It has the rates for residents and non residents
Town of Greenburgh - Parks & Recreation
As far as soccer. Ardsley, Dobbs Ferry and Hastings have a recreational soccer program that is excellent. They also have a travel team. Here is the link to AYSO - they are still accepting registration for next year. My child has been playing since kindergarden and this organization is wonderful
AYSO 324: The Website of AYSO in Ardsley, Dobbs Ferry & Hastings
Unfortunately I can't help you with golf but hopefully the above will help you out. Welcome to Ardsley!