Last summer I went to a conference knowing that I was about a year away from graduating. My boss has prompted to make meaningful connections that might lead to a post-doc, but I quickly fell in with a group of post-docs and young faculty who were sharing gossip and beer in the setting sun. When I mentioned I was supposed to be networking with people who might be in a better position to hire me, and the Aussie (of course) said, "Naw, anybody worth sitting down and having a beer with is worth networking with." He was right, I learned a lot about molecular epidemiology and the tenure track in Europe at that meeting. That genuine chatting with interesting people because you have things to share is the kind of networking I can get behind.
In science, there is sometimes a sense that anyone who would use the word network as a verb is some brown-nosed contact-grubbing social-climber who will ingenuously flatter and hob-nob the right people to get ahead. You hear people saying they wouldn't do that because they want to get by on their science alone. But you don't have to be a sellout to benefit from the knowledge and experience of people you know. Do you really think that being an awesome scientist will help you figure out which jobs to apply for, where you are a good match, or if you are getting the right experience to get there? How to decide where to send your manuscripts, or which grants to apply for? No, that's what your network is for. How many times have you asked how someone got their cool job, to hear that they "just fell into it?" That's code for 'my network hooked me up.' (If you've never asked someone how they got their job, consider that your first Networking Homework Assignment.)
For me, networking has been a lot of friends just letting me know when they hear about jobs opening up or introducing me to people who had success in jobs I am interested in. Or funding sources for training opportunities. Or certification programs. Just telling me things that I don't know about. And listening to what I am hearing too. When I've gone out recently with classmates on the job market, it's part rant, and part checking in- am I doing this right, are you doing what I am doing? Where are you looking for jobs, and are you finding what you want? The job hunt is like a full-time job that I have no training in. I'd be a fool to waste the resources right around me, my friends and colleagues.
How can you be better at networking? People always recommend the book (which I have yet not read) Never Eat Alone. For me, it comes down to two things, making it clear what I need, and connecting other people with what they need. I'm seeing a lot of jobs I have no interest in taking, but I send them to friends I know are also job hunting. And I've tried to tell people I am on the job market, so when they hear about a colleague getting funding for a post-doc or start-ups hiring, they think of me and it saves me time. Maybe that makes me an ingenuous social climber, but my eyes aren't big enough to filter through all the information out there to find only what I am looking for- I'm grateful for my friends who can help me out.