What program is exactly like Sony Vegas but FREE for Windows? Also is it better to edit videos on Linux then Windows?

I'd start here: http://alternativeto.net/searchresult.aspx?search=sony%20vegas

This is an answer to the Linux part of your question. I cannot answer what is free that is exactly like Sony Vegas, although I doubt that such a program exists.

If you are only looking for the best prices for video editing applications, then Linux is definitely for you (they are all free). If stability in your video editing application is more important, then in all honesty (and I am a huge Linux and FOSS supporter) don't switch to Linux if all you want is a cheaper video editor. Video editing has been the bane of many a Linux user for a long time, and while it is getting better all the time, it is nowhere near professional quality at this point.

That being said, there is a few applications that you could look at on the Linux side:

Pitivi or Openshot
These both are very similar in functionality and user interface, and both are quite stable. They are home-user applications, not professional. Think Windows movie maker (well probably somewhat better than that).

Cinelerra or Kdenlive
These are both still more on the hobbyist side of things but much more advanced than Pitivi or Openshot. Kdenlive has more features than Cinelerra but is less stable. Expect Kdenlive to crash on you, ie save often.

Blender
In my mind this is the best and most stable of all, and it is on a professional level. But you might find it difficult to use as it is not just a video editor. In fact most people completely look past this one because Blender is more known as being a 3D modeller and gaming engine, but it does have a fully featured video editor.

On the linux side I use Openshot and KDenLive. I love them both.

Also on the Linux side of things I wanted to add this as well. I remember looking on distrowatch.com. Which is like the top 100 of the 350+ distros available for Linux. One comes to mind which was for audio and video editing specifically in Linux is called Ubuntu Studio. Take a look and see if that is a viable option for you. I think Mac, Windows, and Linux have all strength and weakness in them Some prefer one over the other.

If you want Linux, go Kdenlive. I cannot comment on editing under Windows as I have never done it. But I have never had a single problem with Kdenlive and I love it.

http://www.lightworksbeta.com/ its like final cut pro