How can the puck connect to the internet?

Are you sure you've got the correct pin connected? Maybe you could give me a bit more information about the code you're using and where you connected. Also remember that you always also need to have ground connected as well.

For ESP8266, it's basically what is here: http://www.espruino.com/ESP8266

• Make sure your firmware is up to date
  
• Get an ESP8266 with its own power source (the CR2032 in the Puck won't cut it)
  
• Ensure it's wired up with BOOT/CH_PD as shown in the link above to make sure it boots up in wifi mode
  
• Connect ground -> ground
  
• ESP8266 TX -> D29 (Puck.js RX)
  
• ESP8266 RX -> D28 (Puck.js TX)
  
  

And then this should do it for you:

Serial1.setup(115200, { rx: D29, tx : D28 });
var wifi = require("ESP8266WiFi_0v25").connect(Seri­al2, function(err) {
  if (err) throw err;
  console.log("Connecting to WiFi");
  wifi.connect("WiFi_Name","WPA2_Key", function(err) {
    if (err) throw err;
    console.log("Connected");
    // Now you can do something, like an HTTP request
    require("http").get("http://www.pur3.co.­uk/hello.txt", function(res) {
      console.log("Response: ",res);
      res.on('data', function(d) {
        console.log("--->"+d);
      });
    });
  });
});

Personally I'd also ditch the Puck's 3v battery and connect the 3.3v from the ESP8266's power supply to the Puck instead.