How to travel to other European countries with a Spanish student visa?

You don't need to apply for a Schengen visa. Spain was originally a Schengen country. When you go to Italy from Spain, it's like going to Shanghai from Beijing. There is not even a pass in the middle, and it is not considered to leave the country. Imagine that the whole Schengen area is a big country and each country is a province. It doesn't matter except that you can't go to a small country other than Schengen, such as Britain.