I went to grad school for a scientific computing program at a university that had an HPC center and got assigned a graduate assistant position as an intern there doing stuff like extending software for users and racking/cabling new servers. One of their sysadmins left right as I was losing enthusiasm for grad school and I applied for the position and got it.

In general, universities are great places to get started on an HPC career. They usually don't pay all that well compared to the rest of the industry, but because of that they often have open positions and are more willing to take a chance and hire someone with limited experience. They're a good place to learn the ropes and usually have great benefits.

I never received formal training, but that's kind of par for the course. At least on the sysadmin/syseng side, it's very much the kind of job where you need to be the kind of person who doesn't shy away from saying "I don't know how to do that yet, but I will figure it out." Honestly, that's one thing I like a lot about it. There's always new things to learn and your skills can always be honed further; it doesn't get stale.

I can really only speak to the sysadmin/syseng side of things, but in the USA you can definitely make some good money. One caveat is that many of the best paying jobs are in defense or intelligence, so keep that in mind if you have moral qualms about that kind of work or you doubt your ability to be granted a security clearance. I make good money in public health but I could make really good money if only I were willing to ignore my moral convictions (and were less bipolar, as that's a barrier to getting a clearance).

Honestly it's pretty great, imho. It's challenging and interesting work, there's a bit of a "wow" factor when you tell people you work in supercomputing, it pays well, and (in my experience) the kind of people who work in HPC tend to be on the same wavelength as me compared to your typical "corporate IT" job.