I think attending conferences is really important for students at all levels. I can only afford to send my grad students (though I have sent undergrads to local meetings), but I make a strong effort to send every student who has something they can present to one meeting per year. In practice, this means pretty much all of them, every year. I think this is so important to their development, that this is a priority for me. We also apply for any travel support that is available to facilitate this.
This last conference I attended with some of my students, I noticed that some of them were not really making the most of the (professional) opportunity. Then I remembered that when I was a student (a shy, introverted, not really self-confident student), I spent most conferences hanging out with people I already knew from my undergrad or PhD universities, which was fun, but not really the point. No one ever suggested ways I could get more out of meetings, which would have helped newbie Prodigal a lot. So I decided to start explaining to my students what they should be doing to make the most of a conference (especially since in lean years, we scrape by so I can send them).
Prodigal's rules for conference success:
1. The stated point of a conference is to exchange ideas. So you need to:
Get your work out there! This of course means that it is important to present your work well in a good presentation (THAT DOESN'T GO OVER TIME!) or as a well thought out poster. It means you need to be at your poster for at least half of the poster session. But most importantly, it means you must discuss your work with other people, even outside of your talk/poster.
Attend talks/posters on things that interest you that might not be directly relevant ot your project. This is a great way to get new ideas, and to learn about a new direction/new area. Most speakers will be distilling one or more papers into the allotted speaking time. The good ones will get you up to speed on the key points during their talks, which can save hours of laboring over papers and references. I find that attending talks on topics I am interested in is the best way to spark my creativity with my own work.
Talk to people in your field to hear about negative results. Negative results are usually thrown in to another paper (as an add-on to positive results), so it could be years before anyone knows that a particular line of research is unlikely to be fruitful. People will talk about negative results in person when discussing their research, though. The publication bias towards positive results means that many labs try the same thing that leads nowhere until someone puts it into the literature that the method doesn't work and stops new entrants to the field from trying it. You can avoid that by talking to other people so you don't invest loads of time in something unlikely to be successful. You should do the same for others, by the way, to help save them time too.
2. The actual point of a conference is networking. So there are three things you need to do:
Look over the program and "stalk" people who you'd like to meet. By "stalk", I don't mean actually stalk! Just find out when they will be presenting and attend the session. You can ask a question/introduce yourself at a break. This is your chance to meet people in your field! As a student, it works best with younger, less established scientists, but there are well-established folks who enjoy meeting students. There are also well-established folks who decidedly don't, but that is good information to find out too.
Make sure you meet new people! This may seem obvious, but it is not. It is easiest to do at mixers/coffee breaks, where people are feeling social, or at poster sessions, where there is a natural way to approach people (at posters) and a natural topic (the research). You can also have your existing friends/mentors introduce you to new people. The people you meet at conferences are your peers. They will be reviewing your manuscripts and proposals, inviting people to various events, and possibly telling you about professional opportunities. They will also tell you the real story about their research (the stuff you can't/don't put into papers). You need to get your name out there. Being talented is not enough if no one but you and your advisor know it. You should also follow up with the ones that you clicked best with after the conference via email.
Refresh connections with people you've met before. Ways to do this: attend their talks/posters, contact them and set up a meeting, arrange to bump into them at a coffee break, track them down at conference social events. Even if you meet someone face to face once a year, it is enough to make them a professional contact after a few meetings. It also gives you someone outside your university to hang out with who can introduce you to new people (see previous).
Point 2 on networking (and its importance) came as a surprise to some of my less experienced students. Early on, many of them still believe that science is a strict meritocracy, and are shocked that networking is so important to success. It is best for them to learn this early, so they can start figuring out networking strategies that work for them.
After my students return from a conference, I like to have them give a group meeting on what they saw, with an emphasis on things that will help our projects, things they thought were the most interesting, and people they think we should pay attention to.