When we "buy" something, we pay to gain possession of that item. We can also use "buy" for something intangible.

Example: Buy a dozen eggs. (tangible) / Money cannot buy true love. (intangible) / He was trying to buy her affection. (intangible)

It is an action where an exchange of value is performed. 

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Things are more interesting for "shop". The meaning varies depending on how you use it with other words:  I can go "window shopping" for hours in a shopping mall without buying anything. All I am doing is to browse every store to occupy my time. I can also "shop around" to compare prices and end up not buying anything. I can "shop for" something by searching for a particular product to buy. When I really do "shop", I visit a shop (noun) or website to buy something or make an order by paying.  

Shopping is more of a decision process on whether we should buy or not; buying is the act of deciding to pay so that something becomes ours.