Ok I'll try to be nice for once.
Just like motorcycles, companies make bikes in different engine sizes. They got 250cc, 600, 750, 1000, etc. They also got different companies that make a bike in the same class (engine size).
It's the same with guns. There are many different companies that make different size guns and different size calibers (bullet size). Take Glock for instance, they make a 9mm handgun in 3 different sizes. A full size (5in barrel), compact (4in barrel), and a sub compact (3in barrel), all using a 9mm bullet. It's the same deal with different size bullets.
Glock also makes guns with different caliber bullets (think bikes with different engine size). They got 9mm, .40s&w, .45, 10mm etc etc etc.
Full size polymer framed handguns usually have a 5in barrel
Compacts are usually a 4in barrel
Sub compacts are usually a 3in barrel
With that said, I would not recommend getting a full size for your first gun. You never know, a little down the road and you might think about carrying it and carrying a full size gun is not practical. I'd recommend getting a compact or sub compact.
There are plenty of great guns in your price range. Go hold a bunch and shoot a bunch and get whatever you feel best with.