Even though the show and merchandise are very popular... the most important part is the game itself! The game's goal is to try to become a Pokemon Master and try to catch all 150! However, all 150 Pokemon are not all in one version.
In order to get all 150 you need to trade with someone who has the Blue version if you have the Red and vice versa. Even though that is the main goal, there are other goals you can set for yourself.
Some of these include specify in training a specific type of Pokemon, trying to build a "perfect team", and trying to beef up your Pokemon by getting them to level 100 (the highest they could go) and filling them with vitamins.