What do you mean by better WiFi? Where is it placed in your house? If it's on the 1st floor and you have a weak signal on 2nd floor, you can simply get a range extender, it will boost your signal. And no, there is no monthly fee if you get a router, unless you get a very expensive one and decide to pay monthly instead upfront.
Advantage of WiFi router is the ability to connect wirelessly to the internet. It also provides protection so strangers can't connect to it.