No when your phone connects through wifi it turns off the 3g and uses your already paid for home internet. if you have a download limit with your home internet provider it will detract form your data allowance but in any case its far cheaper than 3g. : )
Many cellphone carriers require you to have a data plan if you purchase a smartphone, whether you plan on using it or not. Using your phone on a WiFi connection will keep you from using up data on your carriers data plan, but you probably can't drop your data plan altogether as most carriers won't let you.