There are two basic techniques in the photo you reference:
First, it uses "clamshell" or "butterfly" lighting — see What is butterfly lighting, and when do I use it? for more. You can easily see this from the highlights in the model's pupils. The resolution is low enough that I can't tell if the fill light (from underneath) is a reflector or an actual light; I suspect in this case that it was actually an additional light.
Second, the all-white background. To do this, you simply make sure your subject is far enough from the background that light doesn't spill, and light it separately so it is completely overexposed. (You don't even need a white wall for this — see How can I get a pure white background in studio photography? for more.)
So: will a basic two-flash setup plus some octoboxes get you started? Yes — you can do this, and get great results. You can even take photos similar to your target. If your softboxes are small, you'll need to get them very close to provide soft light; that presents some challenges and limitations but can still work. I highly recommend making sure that, whatever system you get, your flash power is adjustable from the camera; that makes it so you don't have to fiddle with the flashes directly.
Umbrellas can work if you have space, but in my experience they're less fun to work with than softboxes. However, just one umbrella and a flash? No, you really want more than that in order to get versatile results. (And if you want white backgrounds and versatile results, give strong consideration to three as the minimum.)