From LIC you should cross the Pulaski bridge into Brooklyn, go right at the bottom to Franklin St, and then left and you'll be on the Brooklyn Waterfront greenway. It isn't fast riding though, still through streets. You'll go through one of the more popular spots in Williamsburg, the hipster hangout that is now more wealthy/touristy than hipster. Plus at some point you'll be able to turn left off towards Prospect Park on either Washington Ave or Vanderbilt Ave. These aren't separated bike lanes, but there are a lot of bikes on both these streets, especially Vanderbilt. You can do laps of Prospect (and the old guy on the raw carbon Kuota is probably me) and when you get bored with that you can exit at the bottom of the hill and take Ocean Parkway out to Coney Island. Ocean Parkway has a fully separated bike lane, the oldest in the country, but the intersections make it less than ideal sometimes. Also peds seem to ignore that it is a bike lane.
Queens just isn't a great place to ride, although it has gotten better.
The NYC Century is a great way to see lots of bike routes, as they try to stay on them and hit them all. You will see some bikeways in eastern Queens, but they are fairly far out there and would be otherwise hard to find.