Short answer: yes, they should. It’s very important. But the type of “traveling” that I promote and encourage these days, for graduate students and postdocs (obviously beginning with my own), is not the traveling that I, that most of us did in our graduate student days. There used to be a time when “traveling” for graduate students (in physics anyway) mostly meant going to conferences.
In condensed matter physics, for example, the March meeting of the American Physical Society is by far the most important appointment in North America, and it was long regarded as the one conference that every physics graduate student had to attend.
It’s where most graduate students get to give their first external presentation (albeit only 10-minute long), facing an audience consisting of professors, postdocs and graduate students from other universities, all over the world. There has always been, in my mind, something truly exciting and majestic about that meeting (also referred to as the debutante ball by a more cynical colleague).
I remember the Friday afternoon before the week of the meeting, spent listening to and rehashing all of our presentations, and then meeting almost the entire FSU physics department at the Tallahassee airport, on Saturday morning, all going to the same place .
The attendance (of the order of 6,000) is perhaps not on par with that of comparable meetings in other disciplines, but still rather impressive, and the first time I attended (in New Orleans, in… ouch, 1988… ), was when I first realized how big the field is, how little what I did mattered in the big scheme of things (but for some reason it did not bother me much), and how many there were like me, excited and proud as I was to present their first piece of work, obviously aiming to get to the same place where I wanted to get, eventually. And of course, it was when most of us would see for the first time legendary physicists of whom we had only heard, or read about, until then.
It is in many respects an educational experience, and clearly also a chance for aspiring young physicists to get to be known, starting to establish their own personal networks of connections, meeting potential future employers. Postdoctoral candidates are often informally interviewed at that meeting, whereas faculty candidates try and find out what their chances are at landing a job for which they have been interviewed on campus a few weeks earlier, or of being interviewed for a position .
“Face time” is very limited though. Meeting is very fast paced, lunch breaks are short and days are long, with a lot of parallel sessions and a lot of talks to go to; seldom does anyone have more than half an hour to talk to a colleague. It gets tiring very quickly.
Interesting ideas can be picked up, but mostly by “bits and pieces”. I remember as a graduate student returning overwhelmed, sometimes excited but more often worried about not having accomplished enough in my research work, and not being good enough to make it. Maybe it is a useful reality check, I don’t know.
In any case, things have changed. Travel budgets are meager these days, and it is my sense that most students do not go nearly as often as they did two decades ago.
Smaller conferences, as well as Summer schools, tend to be more effective, perhaps, from the strictly scientific standpoint. They are more focused, more relaxed, it is easier for a student to approach and discuss physics with prominent scientists and/or key players in their research area.
But I think that the most important type of traveling for graduate students consists of actually spending an extended period of time (e.g., Summer) doing their research work at another institution (or a national laboratory), ideally without their PhD advisor there with them.
Possibly it is something to be done only with senior students, who are fairly in control of their research and capable to work rather independently, but I think it is very useful for a graduate student to get an early taste of what postdoctoral life is. How things work when one is on one’s own, how easy or difficult it is to establish quickly a productive research relationship with new people, discussing physics with someone other than one’s PhD advisor — these are all things with which one should better get acquainted before making the move to postdoc.
It is also a very effective way to be known by a possible future postdoctoral advisor. There is nothing like observing someone “in action” in the course of a few weeks, not only directly, but also through the person’s interaction with other graduate students, to make an assessment of his/her suitability as a postdoctoral associate. I do not think that the same information can be gathered in a 30-minute conversation at the APS meeting, or by reading a letter of recommendation.
And I also think it is important to promote as early as possible the notion among graduate students that their profession breaks institutional boundaries, that research work is shared, that graduate students at other universities are colleagues before being competitors (we used to call this “internationalism” — even though that word has political implications that no everyone may be keen on, I think the concept is similar).
The best way to accomplish this, is for a faculty to have this type of arrangement with colleagues at other institutions, with whom ongoing collaborations are entertained. Students can spend Summer, or some other term, doing research work at the colleague’s institution. Financially it is really not that big a commitment, and I think the payoff is far greater than going to conferences.
 Traveling was not as expensive in those days, or perhaps there was more money for these things. Those were still the day when one had to stay on Saturday overnight, if air fare had to be kept low. It was quite normal for a professor to go with his entire group of students and postdocs. Most of us can hardly afford to go themselves now, let alone taking others with them.
 Although I think that, these days, by the time late March comes, for the most part chips have fallen into place, and not much is left to be had.