Controlling an ATmega328P via Wi-Fi
What I would do is start with an Arduino and then once you have that working you can break the circuit down and solder your own with just an Atmega328 and any other parts you need. There's an Ethernet shield for Arduino that might be useful, some more information is at http://arduino.cc/en/Main/ArduinoEthernetShield

Texas Instruments have also released a new chip which costs about $15 called CC3000. This is a self-contained Wifi chip that I haven't used personally but that has some good reviews. There are development boards available that would help. From what I understand you simply give it the name of your Wireless Access Point, the Wifi password and then off you go and you can connect your car to the Internet. The only caveat with this chip is the package - all the soldering pads are on the underside, so you can't solder it with your normal soldering iron. You would have to use a reflow oven or a hot air gun. Some more information on the CC3000 is at http://hackaday.com/2013/06/24/tis-cc3000-wifi-chip-gets-a-library/