If you're on the way to school, you should avoid overspending before you know what's needed in your individual classes. You'll certainly be told what you need as you go.

That said, the nice thing about this style of photography is that it's not very demanding on the camera body itself. You'll want a reasonably large sensor, but, honestly, anything above a point and shoot will do fine, because, as the rest of your question shows you know, it's really about the lighting and studio setup. You'll want an interchangeable lens camera system, either DSLR or mirrorless, because you'll definitely need that flexibility as you learn. And, the camera should have a standard hotshoe, because you'll use that for triggering your remote strobes.

I think there are a lot of interesting things going on in the smaller (but still significant) camera brands like Pentax or Olympus, or in Nikon's new "Nikon 1" mirrorless series, but, again, I think you should hold off buying into anything until you see what your school recommends. It may be that they have a large amount of Canon or Nikon SLR equipment available for loan. So, again, hold off on early decisions.

Now, with all of those caveats about school aside, here's a general outline of a very basic starter kit that you could use to get going yourself:

• Any mirrorless or DSLR camera. Entry-level is fine, and will provide everything you need as well as perfectly excellent image quality. Don't let the review sites trick you into thinking that spending more will make a big difference. However, there are some advantages in going up a level, if you can swing the price increase — see Are there disadvantages to a prosumer camera for a beginner, aside from cost? Really, though, this is not going to be your primary expense, so don't make it into that.
• Lenses in the portrait range. That means somewhere between 75mm and 135mm in 35mm terms; for smaller sensors, convert using the "crop factor". Depending on budget, more lenses means more flexibility. If you're really strapped, and you've selected an APS-C camera body, you could start with just a budget 50mm lens if need be, or even use the kit 18-55mm zoom at its longest setting. It's the fashion these days to use wide aperture to get a really blurry background, but with studio portraiture that's not the only way to do it (I'd often rather see both the ears and tip of the nose in focus). In any case, you'll probably eventually want some different options here, and will likely settle on a few that really fit your working style.
• On a budget, skip the big studio strobes and set yourself up with a radio trigger system for hotshoe flashes. Low-cost means this will require manual power control, but that's okay because you want that anyway. There are a couple of systems which let you set the remote flash power from the controller attached to the camera, and this is a nice convenience (look at the Godox V850 or the Cactus RF60 system as an example).
• You'll want at least two flashes. That gives you a main light and a fill light. Three would be better — main, fill, and an accent option. And four gives you yet more creative possibilities.
• Softboxes. I think the Westcott Rapidbox 2-light kit makes a good start, and will be something you might continue to use. (You might be able to put together a similar setup with cheaper components too; Westcott isn't fancy but also isn't super-cheap. Take this as an example.) These small boxes have the advantage of portability, but, also because of that need to be extremely close to your subject to be effective. They can work to learn the basics, though, and then you can figure out what else you might want in larger and more expensive form. (Umbrellas are a cheap way to get larger light diffusers, but they're kind of a pain. You'll probably want to learn how to work with them at some point in any case, but I don't think it's bad advice to start with these small softboxes instead.)
• One or more of those "5-in-1" collapsible round reflectors. These are handy for all sorts of things — reflecting fill light where you want it, and blocking it off where you need to
• Don't worry about backdrops, as long as you have a moderately-large area to work. It's nice if you have a basically-white wall which you can cover with something black. (Don't worry about making these pure white or pure black — the lighting will do that.)
• Skip gimmicky stuff for modifying on-camera flash. Actually, basically skip anything where the back of your mind suspects that it might be a gimmick. There's a lot of that.

This isn't a "professional" setup in the traditional sense. It's a hobbyist setup that you can use to do very nice work and learn a lot about photography. It might not match what your school suggests, though.

Back-of-the-envelope calculations put the above at about $1400 at the very low end. It'd be nicer if you can double that, with fewer initial limitations on what you can do imposed by your equipment. And of course, there is no top end on what you could spend. It's an expensive hobby or business!

Because photography is an art, there's no inherent reason that you can't make whatever you do have and do into "pro" — except, of course, that it's a competitive and low-margin business with a lot of change and turmoil these days. But that's a separate issue.