The gameplay is simple. You play as black or white. Let's say you are playing black and you have a black disc somewhere on the board, followed by 2 or 3 white discs. If you put another black disc at the other end of the 2 white discs, those white discs become black (yours). At the end of the game, if you have more discs than your opponent, you win!
I do my Othello / Reversi training using software. The top 2 Othello / Reversi software that I use are Magic Reversi and Candlelight reversi. They can let you adjust difficulty settings. Especially Candlelight reversi. It allows you to adjust the computer's mindset to be able to focus on edges, corners, number of pieces and mobility.
I also fight with other Othello players online and the best places to play are at Yahoo Games Reversi, FlyOrDie Reversi and Facebook.