can i use a 700c tire (Gatorskin 700c x23) on a 650 wheel I am thinking of getting some Velocity Rims that are 650c but I am wondering if I will be able to use my 700x23c Gatorskin tires on them? Also what do you think about 650c wheels pros and cons. 

Also, lets say I just buy the 650 rims do I need to change anything else in the bike? I have a Fuji Newest 3.0, I think the 2008 version, if that helps.
No you will not be able to use your 700x23c tire on a 650c rim. The bead seat diameter of 700c rims is 622mm, this will also be the bead diameter of the your 700x23c tire.  The bead seat diameter of a 650c rim is 571mm (see Velocity's Spec section). Your 700x23c tire will be 51mm too wide.