Why do you want to learn Chinese? Your goals inform a lot of the advice. If you desire to work with the PRC, then learning Chinese in Taiwan, which is a very comfortable country for a westerner, will probably greatly harm your prospects and literally put you in harms way if you go to China. People I have met have first hand told me of threats to revoke visas (trapping a person in China) and literally imprisonment for saying the wrong thing, requiring action from the US State department. That was before China went full fascism on Hong Kong.
My low quality advice from someone who has done minimal research but who also has a desire to learn Chinese:
Go to Taiwan and take classes. Learn traditional Mandarin in Taiwan.
Why should you take classes even though you don't want to? Chinese is tonal which is completely different from the languages you already know. I don't know how you will get tonal feedback, training, or practice without classes. The writing system is different than the languages you already know and will likely require different strategies to learn as well.
Additionally Taiwan frequently uses bopomofo rather than pinyin. I am convinced (not through first hand experience) that it is better to attach meaning to completely new symbols rather than bring in legacy phonetic associations from the western alphabets.
The US state department defines Chinese languages as category 4, meaning they expect learning it, on average, to take on the order of 2 years of 30 hour weeks to become business proficient and that is with traditional training.
As far as culture and literature, when I asked a similar question to a well educated Chinese person, they suggested the four classic Chinese novels (https://en.wikipedia.org/wiki/Classic_Chinese_Novels) and specifically said romance of the three kingdoms has become a somewhat apocryphal history of China.