If I put a shield on an Arduino, can I use the Arduino for anything else?
Normally, yes, you may plug other components to your setup after you plug your shield, but it depends on the shield exposing the unused Arduino pins through additional female headers. 

Some shields even provide a little protoboard or perfboard to make it convenient for plugging or soldering additional components. Below are some examples:




Arduino shields are normally designed to be stackable, so you can plug one on top of the other. See more on Arduino shields from this WikiPedia article. See below one example of stacked shields, shamefully stolen from this great answer by RedGrittyBrick.


Photo by John Boxall

But note that you need to study carefully what pins each of the shields use to avoid conflicts between the boards. Such conflicts may lead to damages.