Ok, first of all, you can do either of these first. Install the new I/O plate or put the nipples in the tray/case, it doesn't matter which of those you do first. Make sure you put the nipples all in the correct holes depending on what style case and motherboard you have (e.g.: ATX, Micro ATX, etc.). Consult your case's manual for that. Also, to get out the I/O plate that came with the case, you either push it inwards or outwards. Look at it and see which way. Then just hit it with your fingers and it should fly out. After you know which way it goes out, put the motherboard's I/O plate in and press until it pops into place.
Now insert the motherboard into the case and properly align its screwholes with the nipples you put in earlier. Then simply screw it in and you should be ready to start installing other hardware and cables.
Edit: Oh, and as I said above, don't forget to put washers on the motherboard when you're screwing it in. If a screw makes contact with metal, the PC might not POST.