It has occured to me one possible way to remain invisible to Prism and such is to use a laptop with no personal information on it. You wouldn't of course be able to use email under your own name. But my question is:

Would doing this via a Virtual Computer be adequate or would they pick up the physical IP Address and be able to identify you by your physical computer?

No personal information is needed when they can track your IP address and knock on your door.

Your best bet is to use the Tor bundle https://www.torproject.org/projects/torbrowser.html.en But please make sure to disable javascript, to many flaws. Of course the other alternative would be Tails, https://tails.boum.org/ and that does not come with javascipt disabled.

With all of that the best thing to do is watch what and how you do things. Ex. downloading, using personal emails and so on.

I should have added that this would be done from Hotspots, so the IP Address probably wouldn't identify you. And my question still is, would they be able to see the physical commputer or would they just see the virtual computer?

Remember that MAC addresses don't transfer across route barriers. If someone is listening in on the traffic upstream of you, the will not see machine identifiable information in the raw packets. If the traffic is going to be identifiable, it would be due to patterns of behavior and locations visited rather than tagging the machine itself. That being said, the letter agency you are concerned about already has a program in place to compromise your system. Read the latest Bruce Schneier Cryptogram blog for details on this subject.

You can't realistically prevent these über-funded secret agencies from doing as they damn well please . . . and, until the rule of Law finally begins to impress upon them (which will be difficult, due to the unfathomable amounts of $$MONEY$$ that they spend, and since it is all spent in $$SECRET$$), they remain unstoppable.