How can I tell that my phone uses the Wi-Fi connection for data traffic? I just bought the Samsung Galaxy 2 and my monthly data plan allows me 250MB.

I have Wi-Fi at home with 100GB of data per month.

Obviously, I want to use my home Wi-Fi as much as possible on my phone, and I have no trouble connecting my phone to it.

My question is:
How do I know if my phone is accessing my home Wi-Fi and not my phone company's network when I access the internet on my phone.

I can see that my phone is connected to my home Wi-Fi but it is also connected to my phone company's network at the same time. Is there a way to see when I am using my home Wi-Fi and when I am using my phone company's network?

I am worried that if my phone connects to the phone company's network and not my home Wi-Fi, then I will end up with a huge bill. I just want to make sure I am using my home Wi-Fi when I am accessing the internet on my phone.
If you are connected via Wi-Fi, your phone will only use that connection. The only sort of data that would be required to use mobile data would be MMS (aka multimedia messages) everything else will operate with Wi-Fi if you have it.

One thing you'll also want to look into is the Wi-Fi sleep policy, to ensure you phone never disconnects from Wi-Fi if it is in range.

Go to:  

Settings → Wifi → Advanced (via menu button) → Sleep Policy


Set this to "never", so your phone will always use Wi-Fi (if available) when it's asleep.
 This also saves battery as the distance between your Wi-Fi and your phone is a lot shorter than the phone mast and your phone!

Update
If you are that concerned about using mobile data, search the market for "APN Droid", which is an app that completely turns on/off your mobile data.