As for the best school district outside the city, its really hard to say. First, the best public school districts in the region are outside the city, the city schools can be very good, especially the specialty schools (Bronx Science, Brooklyn Tech, Stuyvensant) but generally are mediocre to bad. The NYC education department is a horrible disaster and reminds us of the days of Tammany Hall.
In westchester, Bronxville, Scarsdale, Chappaqua (Horace Greeley), Armonk (Byram Hills) are some of the best, but there are many more like Edgemont (also in scarsdale), Ardsley, Rye Brook (Blind Brook) just to name a few. I have no knowledge of the other areas you mention but there are a number of online school reports that can give you some data to start with.
Hope that helps its just one of many possible starting points
What are you looking to spend? The best school district for what age?
Forest Hills is great for shcools, but pricey depending on what area.
Queens is great for prices aside from that, but the school district can be an issue depending.
Northern NY is dead on in terms of Mitchell's comments. There are plenty of options out there in Mt Pleasant, Westchester Co., Rockland. The Poconos are a good buy for low price as well.
What about HPD or First time home buyer programs if you truly want to stay in BK?
I have a friend who purchased a home in BK for $250K. 4 BR 2 Bath and NEW CONSTRUCTION!
The details would help me more, but hopefully I am at least pointing you in the right direction.
Let me tell you about Westchester and areas North. Yes, taxes are higher here than in the city. Other than listening to my family complain, I don't know about Long Island property taxes. I live and work in New Rochelle. New Rochelle is a relatively short commute into Grand Central - 30 minutes from New Rochelle's station. The taxes here fluctuate depending upon the neighborhood.
The public schools in Westchester are some of the best in the country. This is a fact that you can research online. Our schools challenge the students in a wide array of subjects and disciplines. This is measured not only by the test scores but by the ongoing projects and performances in the schools as well as college acceptances. Westchester is a large place offering towns for everyone. Further north you get more property for your money than you would in lower Westchester.
Another choice to explore beyond Westchester is Putnam county. Taxes are lower and housing prices are lower as well. You can check out Metro North's site to get an idea of commuting times.