What is pursuing a PhD in climatology like?
There few departments that have "climate" in their name.
Among the closest is the program at Wisconsin from which I attained my doctorate, the Department of Atmospheric and Oceanic Sciences (formerly Meteorology). So there's a strain of physical climatology that emerges from the meteorological and oceanographic traditions. You'll learn more than you want to know about the nuts and bolts of weather prediction, but you'll get a feel for how the physics of the system works.
The other strain of physical climatology emerges from earth science. It is more focused on paleoclimate and thus on how climate changes in nature. There's a lot of very interesting information about past climate buried in rock formations. Emerging from this discipline is an interest in more recent paleoclimate proxies - corals, tree rings, mud deposits, and ice sheets - which contain information on the intermediate time scale (a few thousand years) and which seem to attract more than their share of howling from the sidelines. (People who want to think our current changes are in large part due to nature want to see more variability in the millenial record and less variability in the century record. You may think this is an odd way to approach science, but that's veering off topic.)
Related, and probably the richest climate-related field of study for the mathematically adept nowadays, is computational science. Unfortunately, for institutional reasons the connections between the computational science community and climatology are relatively weak. But if you're very strong both intellectually and in terms of ambition, that's an angle that would be worth pursuing.
This is all what might be called "working group I" stuff. That's the IPCC nomenclature, where there are three working groups. People working on impacts (group II) and on economics (group III) are often broadly called "climatologists", but I am not in a position to help you with this and I assume this is not what you are asking about.
The material is fascinating, but at least when I did this the pedagogy was week. I think the textbooks have improved somewhat, but you'll still have to be something of an intellectual self-starter.
As far as paving the way to a secure job, the importance of the topic is no guarantee.
In the US in particular, one political party is increasingly hostile to climate science as a community. Even if the most extreme elements there do not succeed in shutting it down, the political overlays on the funding make things both unsteady and somewhat ill-organized from the top down. It's certainly not a great situation. Those suggesting the field is rolling in money are simply unfamiliar with how it goes. Funding in the US has been unsteady and not showing long-term growth though it bumps up and down with the political winds.
So as with any career path, much depends on your aptitudes, your interests, and your goals. There are many drawbacks to a career in climate science, including being forced on a daily basis into a conscious awareness of how badly the planet's future is being bungled, which isn't exactly fun.
If you're not much interested in saving the world, it was a much better gig when it was a pure science that nobody cared about. It's not the healthiest scientific community you can imagine, which is understandable given all these pressures.
But if you can associate yourself with a top-rank professor at a good program, all of these concerns will be secondary while you are a student, as there is a lot to learn and it's intrinsically beautiful and satisfying material.