Tajweed is usually introduced as the art of reciting beautifully, which makes it sound like an optional finish applied after the real work is done. It is better understood as the discipline of pronouncing each letter from its correct place, with its correct qualities, for its correct length.
The reason that matters is simple: in Arabic, a small change in articulation is a change in vocabulary. Letters that English speakers hear as one sound are separate letters with separate meanings — the s of seen and the heavier s of sad, the t of ta and the heavier t of tah, the h of ha and the throat h of hha. Swap one for another and you have said a different word. Most of the time the result is nonsense, which is harmless enough. Occasionally it is a real word with an unintended meaning, and that is the case the rules are guarding against.
The rest of tajweed is mostly bookkeeping on top of that foundation. Madd rules tell you how long to hold a vowel, and the length is not decorative — it distinguishes words. The rules of noon and meem with sukoon describe what happens when sounds meet, which is the same kind of assimilation every spoken language has, written down precisely. Qalqalah describes a small bounce on certain letters at the end of a syllable.
None of this needs to be learned as theory first. The usual and better order is to hear a rule, imitate it, apply it in what you are already reading, and only then learn what it is called. Students who learn the terminology ahead of the sound can name rules they cannot produce.
A word on self-study. Tajweed is the part of Quran learning that resists it most, because the whole subject is about distinctions you cannot yet hear. You need someone who can hear them to tell you when you have made one. Recordings help you imitate; they cannot tell you that your qaf is coming out as a k.