Pinyin is the standard romanization in mainland China, not Wade-Giles (which seems to be preferred in Taiwan?Taiwan is in a bit of a confusing place right now with phonetic alphabets for Mandarin. Here's my understanding - I'll use the abbreviations "WG" for Wade-Giles and "HP" for "Hanyu-Pinyin".
- They traditionally used Wade-Giles to romanise things for international consumption. ie Taipei (HP: Taibei), Taichung (HP: Taizhong), Kaohsiung (HP: Gaoxiong) etc.
- They officially switched to Hanyu Pinyin as their official romanisation about 10 years ago, but it seems to only have taken place on a very low scale. Ie the Da'An (WG: Ta'an) distrct in Taipei, or Wuri (WG: Wujih) in Taichung. With regards to people - the second president of the ROC-On-Taiwan is known in English by his WG name Chiang Ching-kuo (HP: Jiang Jing-Guo), while the current president is knowing exclusively by her HP name Tsai Ing-Wen (WG: Zai Ying-Wen). You can see this confusion manifest itself today - we still eat "Peking Duck" (WG), but we refer to the city as "Beijing" (HP).
- Some places in Taiwan are romanized with neither system, like Keelung, which seems to be some ad-hoc romanisation of Taiwanese Hokkien (another Chinese language entirely) that took root in the 19th century. Though interestingly enough, in the most common Taiwanese Hokkien romanisation system it would be "Kelang".
- In terms of education for locals, it's 100% Bopomofo as far as I can tell. I've also yet to meet a Taiwanese person who uses anything other than Bopomofo for input on their phones and computers. Apple atually added an extra tone mark to their Bopomofo IME (high tone is indicated by no tone mark, traditionally), which some Taiwanese people now seem to think is actually part of the alphabet!
- And lastly, in terms of education for foreigners, it seems to be a hugely mixed bag. Many foreigners now demand the use of Pinyin because it uses the Latin alphabet and it's more comfortable for them. A lot of places will still expect you to learn Bopomofo (which IMO has a lot of advantages - it's only 37 characters, stops you thinking in terms of your native pronounciation, and the IMEs for it are way better because they let you filter characters by tone). AFAIK Wade-Giles is dead for teaching, but one famous Taiwanese book for teaching Mandarin I used uses all three phonetic alphabets - WG, HP and Bopomofo, which was quite distracting!
Full disclaimer - my Mandarin is rudimentary and I am not Taiwanese.