I have purchased a new router recently, as the current one in my shared house does not reach all the areas of my floor. So, I thought by buying one of these routers I could connect this to the internet then I would be able to have a better connection. This is the first time I have done this, I presumed it should be straight forward for me to set up as this is what all the reviews of the product told me. 

After opening the box and reading through the manuals, it appears this is not the case?

What I would like to know is, if the internet for the house already has one router connected to it via cable, is it then possible to connect another router? If so, is it possible for this to be connected wirelessly? 
Is it possible to simply plug this new router into the wall and then connect it to an internet source wirelessly then it be free to us?