How do I use an antistatic wrist strap and mat
Actually, connecting to the case is probably better unless you also connect the case to earth ground.  The easiest way to ground the case is to plug it in, but you don't want to do that until you're ready to power on.

What you're trying to prevent is Electrostatic Discharge (ESD), also known as a static spark, from you to the part(s) that you're working with.  Nothing in that equation involves earth ground except that other things that you're likely to touch might be grounded.

So when you're working on stuff, you don't really need to be earth grounded any more than when you're not working on stuff.  You just need to be connected to the stuff that you're working on.

An ESD mat will be conductive enough that you can just set the case on it, put the strap on, and be sufficiently connected to the case.  Then handle all the other parts by whatever it is that will eventually touch the case anyway, and you'll be fine.  No need for earth ground in most cases, though it usually doesn't hurt either.

As a side note, I've built several PC's on carpet without any grounding at all just by handling parts as described above and by casually touching the case a lot.  Never had any problems.  But I'm also naturally paranoid enough in general to remember to do that.