You may wish to consider an ESP8266 board ( Adafruit Feather HUZZAH, NodeMCU, WeMos D1 mini etc) rather than an Arduino+shield as these have inbuilt WiFi connectivity. Searching for "esp8266 openweathermap.org" will show a large number of projects to get you started.
If you have a slightly higher budget you may look at a Raspberry Pi or Beaglebone board as these have significantly more memory & processing power and more connection options.