I have a new strat too and when i got it it went out of tune when you looked at it!
Heres what i did.
1, Make sure the trem is parallel to the body, that is make the trem studs the same height and adjust the saddles for the radius.
2, If you want the trem to float make it parallel to the body
3, To make the trem sit at the right height you may have to adjust the micro tilt on the neck so that there is a little neck angle.
4, cut the nut slots to the proper height and the double over some 1200 grit paper to polish the slots, then put some graphite in there from a pencil.
5, This is all i done and now it just doesnt go out and i mean it doesnt. All these thing need to be done or it will go out, I did not change the tuners, nut or string trees
EDIT
Put 2 springs instead of three and tighten them up, it feels much better too. I use 10's BTW
The American strat trem is really a good trem if it set up right. They never are out the box.