Simple ways to connect with others:
- Via the wireless connection system. You don't need any plugins or extentions for this - so long as wireless is allowed (Settings @ DSi/3DS). This allows you to fight/trade with players in a short range of distance.
- Via the Internet. Wi-Fi or other wireless connection methods are required for this to work which can all be set up in Settings. This allows you to trade/battle globally, with a compatible modem/router. You can connect with people on the other side of the world with this.
See my comment above for your second question. It isn't allowed here so it's not in the answer, but a comment.