That depends (the usual answer, I know 😉 ). If your "Internet cord" is the coax cable then you can move your router only to place where you have coax ("cable") outlets.
My personal take on router placement in a house: if you have a place where all the house-internal Ethernet cables come together, chances are that's where your cable comes in and is splitted too. Use that place. Forget the WiFi of the router and use the Ethernet cabling to install APs in the various places. You will have an easier setup life than finding the "perfect" spot in your house.
For the "AP": look for Wireless Extenders and check they have also an AP functionality. To my knowledge Netgear has it, all my Dlink Extenders I used so far have it, Linksys has it for certain models (search for "Linksys extender as AP" or such). No idea about other models/makers.