The first use of the Biplane tool is free (and do note: this is for the tool, so buying another plane does not get you another free flight!). After that it will cost FarmVille Cash to operate it. Cost to use the Biplane depends both on the number of plots and how long the crop still has to grow. The first night of release, it was noted some users weren't receiving the free use after purchasing it. This has been fixed.
As of November 12, 2010, Farmers' can use the Biplane to instantly grow their trees.
As of July 28th, 2011, Farmers' can use the Biplane to instantly ready their animals.
Even without the biplane you can now instantly grow your crops.