[Anguirus]

^ Possibly Godzilla vs. Mechagodzilla 2. Made in 1993, got a US DVD release around 2004.

If you're 17 and you like this genre it is absolutely required to see the Japanese cut of the first Godzilla. There is no genre without that. And standing all on it's own it is a fantastic sci-fi movie and movie in general, which is not true of most of its legion of imitators.

I'll make a few more recommendations from the list of Godzilla films readily available on DVD in the states.
General high quality: Mothra vs. Godzilla, Godzilla, Mothra, King Ghidorah: Giant Monsters All-Out Attack
Old-school cheese: Monster Zero, Destroy All Monsters, Godzilla vs. Mechagodzilla
New-school cheese: Godzilla: Final Wars

Also, Netflix Streaming is trickling out Godzilla films from the Classic Media DVD collection, including the 1954 original cut of Godzilla.