As a retired New York City area realtor and long time Nassau County resident, allow me to educate you on some misunderstandings in your question.
Valley Stream is located in Nassau County - not Queens. Valley Stream is not part of New York City. Therefore, since Valley Stream is a suburb of New York City, housing is expensive and so are property taxes. Currently, a 3 bedroom rental averages $2,500/month. and up. A 2 bedroom rental averages $1,800./month and up. Beware, the size of rooms will be smaller than what you are used to in Florida.
If your budget is up to $1800., you need to expand your search area to nearby communities - Elmont, Franklin Square, Floral Park, Malverne, West Hempstead, Rockville Centre, etc. These towns should be cheaper than Valley Stream. If you want to live within New York City (Queens), then look in Laurelton, Rosedale, Cambria Heights, St. Albans, Springfield Gardens, etc. Rentals average $2,000. and up.
For New York City (Queens), use http://www.streeteasy.com. This is a real estate website exclusive for NYC listings. It does list nearby Nassau County towns but it's more for Queens. For Nassau County, use Zillow and Trulia. Your best method is to contact a realtor and have them email you listings directly from the MLS.
Beware, the cost of living is very expensive. If possible, try to have your employer increase your income to at least $150 - 200,000. Below $125,000. means you will struggle financially and with 3 kids, it will be difficult.