Using the Spica for the first time. How do I know if I am off the internet after using the phone to browse?
You can look at the signal bar, the signal bar would be grayscale when it's not connected to the internet, green if it's connected, and have arrows when actively transmitting data.
The phone will only connect when it needs to, such as when you're browsing or when an app needs to check for update. Unless you're charged by connection time (I don't know any carrier that does that, most critters charge by amount of transmitted data), there's no need to disable them manually when you're not using it. If you need to disable Mobile Data though, you can do that in the Settings > Wireless & Network.
I would suggest auto disabling WiFi when your phone is locked. This will not only make sure no data is being transmitted, but also save a LOT of battery. Im sure there are many ways of going about this, but this is the easiest and most reliable method that I know of.
Also, here is how to tell your WiFi status:
If you just follow the instructions I've provided below, you should have satisfying results in just a few minutes!
Good luck and I hope this helps you!