October and November are the best times to try and find an apartment in Flushing. These months are best because they strike a balance. The summer time has more listings but is competitive, and the wintertime tends to not have enough availability. October and November provide a nice middle ground.
Much of the wisdom that applies to the rest of New York City applies to Flushing as well. Foremost amongst the tips you might get is to not be afraid of using an agent. It's true, you might end up paying them some fees. But what they will save you in time, effort, and potential run-ins with unscrupulous landlords is usually well worth the hassle.
The people of Flushing have perfect access to the best public transportation system in the country. The New York City subway line has a dedicated Flushing line that runs from Times Square right to the Flushing-Main St. Station. Once you're on the subway, you easily access anywhere in the city.
Visiting the Flushing Chinatown will give you access to a lot of fun shopping and great dining options. One of the things to not miss in Flushing is Citi Field, where the New York Mets play, and is also home to a Mets museum. For music fans, a visit to the Louis Armstrong House Museum is highly recommended. And of course, if you run out of things to do right in Flushing, you're only a subway ride away from the entertainment options that await you in the rest of NYC.