What are some good ways to figure out one's interests in mathematics, before applying to PhD programs?

Are there mathematics seminars to attend in the summer, or do they typically only occur during the academic year?

I think none of the answers thus far actually addressed this. At most US universities, seminars don't happen, at least not regularly, during the summer. One reason is many faculty will be traveling. However, sometimes some people, possibly grad students, will organize an informal seminar to read through some books or papers. If other people aren't doing it, you can ask around to see if people are interested and organize it yourself.

What else could I do?

In place of seminars in the summer, there are usually lots of conferences and workshops as well as summer schools specifically geared towards young people. Look around for what's available (you can try asking some professors if they have suggestions) and go to what you can. Many of these have funding for students, but unfortunately most of the funding may have already been given away by now.

Edit: By the way, it's not expected that you know what you want to research when you enter PhD programs (in math in the US, say). Many people who start aren't even sure if research is for them. Most programs are designed so that you can use the first couple of years figuring out what you want to do. (And even later, research interests may change---I may be doing something completely different in a year or two from what I'm doing now.)

Try reading the Princeton Companion to Mathematics. The editor insisted that the articles were actually comprehensible!

The fundamental problem in selecting research areas is that how an area looks from the inside once you actually know it, is very different to how looks from the outside. This is particularly the case when people tell you about the applications or nice pictures -- these really tell you nothing about what doing research in the area is like.

What sort of problems do you like to solve? What sort of techniques do you like to choose? can you see yourself fiddling with an integral all day? do you like abstract nonsense or prefer things more concrete?

In pure math, research seminar talks are notoriously specialized, and not always well done. To give you an idea, I'm a postdoc doing research in PDE, and I've gone to many a PDE seminar that left me in the dark after the first slide, because the subject matter was too far from my own corner of PDE, or because the speaker did a bad job of introducing his work to non-specialists, or a combination of the two. At the very least, as someone who's "looking to get into PDE", you're very far from being in the target audience of these seminars.

You're better off looking at graduate-level textbooks in various subjects. This isn't really a window into the research world, but it's likely the best thing you can do right now to get a sense of what you like. For PDE, Evans's book is a standard choice.