One reason beginners get overwhelmed is learning tajwid in a random order — memorising terms before the foundation is there. Each stage really does build on the one before it. Here's the sequence that makes the most sense, from the most fundamental to the more detailed.
Before any rule, make sure each letter comes out of the right place. The makhraj (articulation point) and the letter's qualities (heavy/light, hissing, bouncing) are the foundation — without them, the rules built on top mean much less.
Next, the length of each sound. Master Mad Thabi'i first (2 counts), because it's the root of every other Mad — wajib, jaiz, 'aridh, lazim and the rest are just variations in length and cause.
The most productive chapter to learn, simply because it appears so often. Its five rules — Izhar, Idgham (with and without ghunnah), Iqlab, Ikhfa — become easy to spot once you know which letters belong to which group. Start at Nun Sukun & Tanwin.
Only three rules and few letters, so it goes quickly. Mim Sukun completes the "silent letter meets the next letter" pattern you already learned with nun sukun.
The remaining chapters are relatively short: Lam Ta'rif (Qamariyah & Syamsiyah), Qalqalah & Ghunnah, and Tafkhim & Tarqiq. By this stage you're mostly sharpening your eye for patterns you already recognise.
The last stage isn't a new chapter — it's application: reading a whole surah while applying every rule at once. Start with the short, familiar surahs of juz 'amma, slowly, noticing each rule as it appears.
Ideally yes. Makhraj and letter qualities are the foundation; rules like ikhfa or qalqalah only really pay off once each letter is coming from the right place. That said, the two can be learned side by side.
Because Mad Thabi'i (2 counts) is the root of nearly every other mad. Once you have it, mad wajib, jaiz, 'aridh and lazim are just variations in length and in what causes the lengthening.