The most common method of wireless communication with Arduinos is with Xbees (basically wireless serial port modules). It is relatively easy to set up and gives about 300 feet line of sight range or 100 feet obstructed range. There is also bluetooth and wifi ethernet available as options as well.
If you really want to have an internet accessible robot then a wifi module would be the appropriate route to go. There are libraries readily available to turn an Arduino into a web server.