Emailing a contact you have never interacted with would be the least intrusive channel of communication. You can introduce yourself via email and suggest a time to connect on a more personal level, such as a phone call.
Many people who network never actually follow up after getting a business card.
Be sure to include how you received the individual's information and feel free to suggest a phone conversation if feasible with the contact. The key is to be respectful of the person's time and show appreciation for any support they can provide in advance.
Treat them like an interview in the sense that you want to present your best self to them. They may have a job opening or know someone who has a job opening.
You know they are going to ask you about school and how things are going, so have your elevator pitch ready to go when they ask.
I would advise being versed on what it is you are actually looking for and how you might best leverage that contact/resource.
One way to stand out though is to not just send a note asking the contacts to help you to find an opportunity at the organizations they represent without actually doing some research and applying to positions of interest.
If you are the one who requested the meeting you should offer to pay. However, also be humble enough to accept the other person paying.
I would not go into the conversation immediately wanting to discuss your future career. You should let the conversation flow naturally. More often than not they will ask you about school and what your thoughts are about the next steps once you graduate.
Send a thank you note to each person that you meet with and follow-up with them occasionally to keep the relationship there. You never know what may happen in the future.