We are making a robot and one of our tasks is to detect arrows and follow them to find the goal. I need to know how to detect those arrows and what are the sensors that we can use to achieve this?

Thanks in advance.

 

For example a web camera could do it.

 

You could use a camera if you have adequate processing power to "recognize" the arrows as visual items and ignore others. This is rather easy these days with available open source software.

Are the arrows made of a particular material that can be detected? Metal detector or such?

You may want to do some research on "line-following robots". This is a standard project for robotics classes these days, and there is lots of information on them.