I am a student of mathematics and physics but have been inspired to learn more about linguistics after having learned a new language this year. I found the subject deeply fascinating. As of now I am a bit blind about how to approach in learning the field. I would like some advice on a general progression of learning linguistics. Does anyone have any insight on a logical way to learn linguistics?
Linguistics is different from physics and math, where there is a well-defined curriculum for the first year, then the second year. One (non)-answer to the question is "What are you interested in in linguistics?". Then depending on your answer, people will suggest reading X, versus Y, or Z. For example some people are fascinated with regional dialects of English, so someone might suggest reading a sociolinguistics book (especially by Peter Trudgill).
However, there is a "classical" answer to what constitutes the basic foundation of linguistics as a technical discipline, namely phonetics, phonology, morphology, syntax, historical linguistics and (if you can find good materials) semantics. The typical. I have used Linguistics: An Introduction to Language and Communication by Akmadjian, Demers, Farmer & Harnish at the most basic level (that was an earlier edition, never saw the current one). This gives an elementary introduction to the questions that the main fields of linguistics address. There is a question as to what's the best order to present topics in. My favorite is syntax, semantics, morphology, phonetics, phonology, historical (which is not the ADFH order). My rationale is that it is much easier to show that there are significant scientific puzzles to solve in syntax, coming from English (many people hate learning about languages that they don't know, and it is extremely difficult to demonstrate the basic problems of phonology using English, though there is a little bit that isn't trivial or hyper-controversial).
If you are still interested after that, there are introductory textbooks for each of these sub-areas, indeed a series from Cambridge (the "Introducing" series: lacking a historical textbook). Or, there are other textbooks that focus on each of the specific sub-disciplines.
If you have that knowledge as background, I would then suggest attempting to undertake some form of original research in an area that interests you. I don't mean that you should plan to actually publish a research paper in a peer-reviewed journal, I simply mean, do a serious project. It is easiest to do this if you can come up with a data domain, for example conducting an experiment on vowel formant trajectories in a particular context of English – or, anything on some less well studied language. The point is that you learn by doing much more than you learn by just reading.
Linguistics is such a broad field, that any introduction will inevitably leave gigantic gaps, and there are so many different ways to structure a linguistics course.
One way is to choose a few "core subfields" and look into those. At the university I went to, the 3 core linguistics classes were "Phonetics and Phonology" (sounds of language and how they are used), "Syntax" (how words and phrases are combined to make utterances people use), and "Semantics and Pragmatics" (how utterances get meaning). Other places might place more emphasis on "Historical linguistics" (how have languages changed, how do we know about languages of the past), "Morphology" (how are words formed), "Sociolinguistics" (language in the context of who uses it), "Psycholinguistics" (how people's brains process language), or "Computational Linguistics" (how can people formalize language into something that can be processed or produced)
Another is to find an intro textbook, or find a syllabus for an intro to linguistics class and read all the books cited on it.
If you like learning from videos, you might want to check out the Ling Space, a YouTube channel giving overviews of various areas of Linguistics.
Yet another approach is to find a language or language area you are interested in, and trying to use that as your basis. A grammar of a language or a survey of languages of a region will probably have all the "basic parts" that an intro ling class would split the discipline of linguistics into, but giving more concrete examples for each.
Some people like to approach linguistics from the angle of field work (how to document a language), and if you want to try that way, you might want to read a field methods guide or handbook of phenomena that a field worker might use, like Describing Morphosyntax by Thomas Payne.
Coming from Maths and Physics, the most important intersection with Linguistics would be Logic, often treated in Semantics class, but inseperable from Syntax and Philosophy and informations science. Assuming a good background in Programming for applied sciences, a basis in theoretical Computer Science with regards to the hierarchy of formal grammars (Chomsky hierarchy) should also be of interest. If 6726 already pointed out what compromises the fundamental aspects onto language, these surrounding tangents should be noted, if it may offer a smooth transgression into the field of linguistics, which endevors many crossovers with other fields, i.e. historical linguistics, socio-linguistics, philosophy of language and cetera, to understand that the fundamental aspects are mere tools to work in these universal problems, or at least, if those broader fields can illuminate the make-up of the fundamentals.