You need to turn on Wi-Fi and input your router password if you have one. From there it will save the password and all you need to do is turn Wi-Fi on when you want to use it. It should automatically seek it out and log into it.
You have to do the initial authentication -- picking the WiFi network you want to connect to and entering the password -- but from then on, whenever you are in range of your wireless network, your phone should automatically connect.
All as above. Even when connected, it is normal to get a few MB of cellular data use per day as the phone checks in to home base and does some housekeeping chores. The only way to fully prevent this is to turn off cellular data when at home. For me it is not worth it, but for others it may be.