Yes, you can hook a wireless router up to your cable router/modem. Yes that will enable you to have cable internet both wired and wireless, at the same time, at no extra charge - after buying the wireless router. No, you will not need to unhook/unplug anything after things are setup for different computers to access your cable network.
You need to buy a wireless router. You hook the network cable from the router/modem provided by the cable company to the "in" port on the wireless router. You hook your network wire from your computer to one of the "outs" on the wireless router. Now your computer can access the internet and so can those with a wireless card.
You may want someone a little computer savvy set it up for you, so your network will have security over the wireless network - to keep your neighbors from using "your internet" and keep the information on your computer safe.