81dojo is the main site for polite and proper shogi, where many players may wish to discuss and review with you after the game is over. Players are expected to greet each other before games and resign if lost. 81dojo allows you to play illegal moves (and therefore lose immediately). This is the only site I'd use for long time controls.
There's also the Shogi Wars app, which is meant for very quick games, no chat, over and done with. You can either subscribe or play up to 2 free games a day. It's more graphics-oriented, with good AI and statistical analysis of style, accuracy, etc.. Official app approved.
Shogi should be included under the chess variants. I can't find a site anywhere that will allow me to play shogi like we have for chess. Is there anybody that would want to play shogi too?