How to network for spring internship?

I've never networked before and not sure how I would use networking to get a spring/summer internship. I have a free spring quarter at school (April-June) during which I'm hoping to get an internship (if something like this exists), so I would probably have to do this through networking since usually they only recruit for summer interns. I was going to use my school's alumni database and contact people, but I'm not sure how to proceed.

When and how do you let them know you are looking for a "spring internship"? Do you schedule an informational phone interview with them and then wait for them to make the first step in asking you if you are looking for an internship, and also wait for them to offer to forward your resume along? Or do you just say at some point in your conversation that you are looking and ask if they would be able to help? Since I will be calling people I don't know, is it appropriate to ask for help on the first call or do you need to know them better?

Thanks in advance for any responses!