On the tire shine (I use the same stuff, it works well), just let it dry and buff off the excess, so you won't get any slinging on the body panels.
Basically, with the clay, you can remove the majority of all the crap that has bonded to the surface or won't come off simply with a wash. You'd be surprised seeing all the stuff they pick up that you can hardly tell is there. This should tell you all you really need to know about clay barring:
Detailing Clay & Pre-wax Cleaning