Mandarin has exactly one sound system to learn - 21 initials, 3 medials, 13 finals and the tones - and two alphabets to write it with. Choosing between pinyin (拼音) and zhuyin (注音, a.k.a. bopomofo) is a geography decision more than a learning decision.
What each system is
Pinyin (1958, mainland) writes sounds in Latin letters: 你好 → nǐ hǎo. It is the world standard - textbooks, dictionaries, keyboards, street signs in the PRC. Zhuyin (1918, standardized in Taiwan) writes the same sounds in 37 purpose-built symbols derived from character fragments: 你好 → ㄋㄧˇ ㄏㄠˇ. Every Taiwanese textbook, children's book gloss and dictionary entry uses it, and it is how Taiwanese people type.
The real differences
Same inventory, different alphabet - the mapping is nearly 1:1 (ㄅ=p, ㄓ=zh, ㄧㄤ=i ang). Two practical differences are real. Zhuyin marks syllable boundaries unambiguously - nothing like pinyin's xi'an/Xian ambiguity - and Taiwanese input methods are built on it, so knowing zhuyin means typing the way locals do. Pinyin doubles as your keyboard input on any device worldwide, which is a big practical edge outside Taiwan. Both mark tones; zhuyin puts the mark at the end (ㄏㄠˇ), pinyin on the vowel (hǎo).
Which to learn first
Mainland track: pinyin, non-negotiable. Taiwan track: zhuyin is the better investment - you will meet it everywhere, and Taiwanese friends will correct your pronunciation in zhuyin. The second system takes a weekend: the 37-phoneme chart shows every symbol in both systems with real character audio in both accents, so converting is a lookup exercise, not a new subject.
Common questions
Is zhuyin harder than pinyin?
37 new symbols vs letters you already know - so the first week is slower. After that the difficulty is identical; some learners find zhuyin cleaner because symbols cannot be misread as English.
Do Taiwanese people use pinyin?
A variant (Tongyong/Hanyu pinyin) appears on some signage, but daily Taiwan runs on zhuyin - textbooks, dictionaries, and the standard phone keyboard.
Which is better for learning tones?
Neither affects tone learning - tones live in the sound, not the alphabet. What drills them is minimal-pair listening and shadowing, which is why the chart plays every phoneme with real audio.