This article is from the Final-Fantasy and other Square Soft Games FAQ, by firstname.lastname@example.org (Nick Zitzmann) with numerous contributions by others.
Basically, the storyline is very simple. It is about a boy and his dog
(whom the player names). from Podunk, USA. The boy is a bit obsessed
with Science Fiction films, as he is constantly referring occurrences in
the game to the films. The game starts with him and his dog leaving a
movie. His dog runs off to chase after a cat, which leads him to a huge
mansion. The dog runs in the mansion, and the boy follows. Inside, they
find a huge machine. The dog chews on the wire, and the machine sucks
him up inside. The boy has no choice but to follow. They find themselves
in a spaceship. The professor who used to live in Podunk in the 1960s
(it was his mansion) throws the boy and his dogs to some sentries, who
are easily beaten with a bazooka found there. Then, the boy and his dog
jump out the spaceship and fall to the earth. The boy also finds that
his dog has mutated.
The game play is similar to that of Secret of Mana. However, the magic
is a bit different. You use something called "alchemy" if you have ever
played the Ultima Series, the magic system will be VERY familiar to you.
Using the L and R buttons, the dog can "sniff" out items you need to
cast a spell. I.E. ash, crystal, water. Then, to cast a spell, you need
like 2 ash and 1 water.