You will start out with one of three Pokemon; a Bublasaur, a Charmander, or a Squirtle. You must train your Pokemon to fight other Pokemon and weaken them. Once a wild Pokemon has been weakened enough, you can capture it within a Pokeball. If the Pokemon is still strong enough, it will break out of the ball, but if you wear it down until it faints, you won't be able to put it into a Pokeball. You can only carry six Pokeballs and monsters at one time, but you can store the extras at Pokemon Training Centers and PCs (or Pokemon Computers).
From time to time you will run across rival trainers, who will challenge you to matches. They usually hang around Gyms, but you will run into them in other places. You won't win your opponent's Pokemon, but you will gain valuable experience and some cash, which you can use to buy items for your Pokemon.
Trading is key strategy in Pokemon. A Pokemon you got from another trainer will evolve faster than one you caught in the wild. If you can, strike a deal with a fellow player, so he or she will take your Pokemon for a while and build up their levels, then trade them back. But he or she may want a huge favor in exchange!
If you happen to have a Japanese version of Pokemon, you can't trade any Pokemon from that version to the US version. If you do, the monsters will be messed up, some may not gain experience, and you stand a good chance of losing your saved game.