The best contacts for you to have are people in your field or the field you want to be working in. Where can you find a bunch of people who are all in the same field and interested in meeting new people? A professional organization.
There are many, many different types of organizations, associations, and other gatherings of like-minded professionals, and there may just be one near you. They're usually centered around cities, so if you're not living in a city you may have the most luck searching for a group in the city closest to you, or searching for a state organization that meets in different locations throughout the state.
Search the Internet for "[City] [Profession] Association." Try substituting other words for "Association" like "Organization," "Guild," or "Society." You may also have luck searching Facebook or Meetup.com to find groups who use these sites rather than their own websites. Meetup even has a subcategory called "Professional Networking Groups" where you can find all professional networking organizations in your area that use Meetup.
Some organizations, particularly the more established ones, have a membership fee to join, but some don't. They may also have a greatly reduced fee for students. Some require a membership to access their job boards but not to attend their events, or vice versa. Many organizations have both structured, regular meetings and low-key happy hours or other socials, so you can determine which environment is more comfortable for you.
Professional organizations are a great way to meet people who are excited about the same things you're excited about and want to talk about them. And that's a great way to make a new connection!