Is this a good idea? How do professionals in academia and PhD students organise their notes?
Personally, I think this is a great idea. More routes for the sharing of knowledge do good for society as a whole. Often, notes taken by students become the definitive resource for other students.
Academics who have extensive material typed up in notes tend to present it in a low-tech way. In lieu of elaborate web design or a flashy interface, there's usually just a list of PDF files written in LaTeX, often on a site made from bare HTML. Here are some physics examples for inspiration:
And for math:
All of these notes were taken as undergrad or grad students.
Also, should I keep my notes publicly available?
I don't see why not. There are occasionally people who express fears that, if it becomes known that they made a single typo while an undergrad, their scientific career will be over. This is ridiculous!
Most people who see your notes won't read them, most of those who read them won't do so in enough detail to spot typos, most of those who spot typos won't be evaluating your career, and in the absurdly tiny chance that a professor thought it was worth their time to comb your personal notes for mistakes, any they found would be completely understandable.
Edit: I feel like I should address some of the objections posted.
The first is the notion that a student will just be adding "noise" unless their lecture notes are better than all others available. This is an impossible standard to meet, and not a useful one. More takes on a given subject can be useful, because they'll be pitched at different levels and from different perspectives. To see how unrealistic this standard is, note that you could apply the exact same reasoning to professors: why should they be permitted to teach anything at all, when better, publicly available notes, lecture videos, and books by other professors certainly already exist?
The second is that public notes could reveal secret information of military value, such as on "rocket navigation". While it's certainly true that secret information exists, it obviously isn't going to be taught in university classes!
The third is, in full, "Who are you?" I don't understand the content of this objection. Who are you?
The fourth is that taking notes could be stealing intellectual property. Of course, this could be a real possibility depending on the context, but in the fields I'm familiar with, it would be implausible. For example, suppose you learned Einstein's "train" thought experiment from a professor, establishing the relativity of simultaneity. There is no reasonable way one can call this the "property" of the professor. It's a simple and beautiful argument that has been passed down by oral tradition essentially unchanged since Einstein gave it. Again, one can see how ridiculous this argument is by repeating it: if you're stealing from the professor by learning and repeating the argument, then the professor is stealing from whoever taught them in undergrad! The only person one could plausibly say "owns" Einstein's thought experiment is Einstein himself, and we do acknowledge that ownership, because we refer to it by his name.
In fact, usually a professor's own lecture notes will themselves be "stolen" from a variety of sources. For example, they could use notes from previous versions of the course as a base. Very often, they will follow an existing textbook and only slightly tweak the presentation. This is fine: if everybody lecturing on relativity had to create their own novel explanations from scratch, universities would grind to a halt -- and the explanations would be much worse, too.
Admittedly, the concept of intellectual property could apply to specific novel things. For example, the professor may draw a diagram in a particular way, so perhaps a lawyer could say the professor "owns" the reproductions the students make. I've never seen a case where legal force has been applied this way. Feynman didn't copyright his diagrams, he showed their utility by demonstration. He gained in reputation by people learning and using his methods, not by suing anybody who tried.