Sunnyside is a bit of hike to City Island + tolls.
JimsCal gave you the link for the Steppingstone facility. They will accept non-Great Neck residents for a slightly higher fee (not sure what it is now). We kept our boat there for 2 seasons and it was a positive experience over all. They do have bathrooms and a snack bar but the rest of the park is for Great Neck residents. Launch service on weekends and otherwise you can dinghy out to your boat. If you do go here make sure you have 2 pennants from your mooring - don't ask how I know this.
There is also a marina near LaGuardia and the fair grounds. It is a bit industrial compared to Steppingstone (SS) and is firmly in the East River while SS is in the westernmost LI Sound with better sailing right there. SS does not provide winter storage for boats (only summer) so you will have to shell out another $1k+ for winter storage somewhere else like City Island.
The next best option (in my opinion) would be the Hempstead Harbor Club which is in Glen Cove. It is a bit of a drive to get there but the Sound is also wider and more open there. HHC is also reasonably priced, has nice members AND has options for winter storage right there. There are some members of HHC on this forum (JimsCal may be a member there for all I know).
In the summer you will probably get to know most of route 25A as the LIE can be a parking lot, especially on Sunday evenings as you probably already know.
People also do put moorings in Little Neck Bay but I know precious little about what costs or clubs are involved with that. I'm also of the opinion that Little Neck Bay is pretty damn shallow and muddy but have no first hand experience to state this as fact. I'd suggest you think about getting a Maptech chart book for the LI Sound which will show you all the marine/marina facilities up and down the Sound. There is also ActiveCaptain which is free if you register to use it that gives similar information.
Hi, Thanks for all the replies. We are in Sunnyside Queens, I drove out to City Island this weekend. Did not realise that it would be such a hike and I don't think it's a feasable place to moor for us. I've never heard of Sreppingstone/Kings Point but I just looked on the google map - looks good. Is there a particular marina there you would recommend ?