How do you build connections in your 20s?

Just started working for a respectable BB (in ER, not IB) and realized how important connections with companies and clients are in moving up the ranks. Obviously, it's not the only factor, but I think it's one of the big ones. How would someone in their early 20's go about building connections? While I do have my fair share of time meeting people outside the office for business, most of the time is still spent in front of the computer. Any ideas or tips? Or is this something that just comes along later on in my career?

Note that I'm not asking for tips on how to present myself to build connections (that's a different topic altogether). My question is more how should us young guys build our network given the time constraint and the fact that the big guns out there already have a solid network of their own.