Did you disable data through Verizon? That is likely why it still visits websites, because you are accessing them through the 3G network (probably on accident, too). I would disable data as soon as you can (through My Verizon online, in-stores, or over the phone), because it is charged at a ridiculous rate of $1.99/MB. If you do it correctly, you can still send picture/multimedia messages while still having data blocked (read around the forums for exact details how to do so).
If you have disabled data, it could be that only a few websites are 'opening', and only ones that you've been to before. On my phone, the google homepage always opens, but that is because the page data is stored in the phone's memory cache. And yes, the phone never truly 'exits' the browser, it will stay on the last page you opened. Whenever i quit the browser, i just click on one of my favorited sites and treat it as a homepage. I also turn wifi off whenever i'm not using it, otherwise it would drain the batteries really quickly.
