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Pardon the apparent negativity but the harsh truth about socializing in Taiwan may be boiled down to "friendships in Taiwan are mostly a means to an end" or "a friend in need is a friend indeed" or perhaps "only fair weather friends exist in Taiwan."
I know a 20ish Brit English teacher who bought a Tiffany diamond ring and sent out invitations to mom and relatives in the UK to have his Taiwanese fiancée cancel the wedding at the 11th hour. He doesn't know what happened but you can bet your bottom dollar it's due to his "career prospects" and not owning a US$1 million condo in Taipei.
If a Taiwanese "friend" calls to meet you, the chances are he or she wants something such as to pitch an MLM product and membership.
Non-Mandarin speakers won't know it but this popular TV show in Taipei had a host who runs a pawnshop appraise jewelry and collectibles brought to the program. He exposed how many Taiwanese have been defrauded by "neighborly" jewelry stores who hyped the value of jewelry that had been doctored.
With "friends" like that, who needs enemies?
Not many Taiwanese would admit it, but "friends" ripping off each other on this island is a time-honored tradition.
In fact just check how "friendly" many Taiwanese bosses treat staffers. If only these workers dare to talk openly.

Sure, you can expect some cultural and language issues in Taiwan, but socializing and intergrating into the local community is no different than other countries. Join a club such as Toastmasters. Due to your native speaking abilities, you can become popular easily and being looked up to. You can also join activity groups or sports groups where you can meet like minded individuals. Volunteer to local services especially those which requires a foreign language. Give generously to your community and you will receive much more in return.

Your advice sounds good in theory. I've been to Toastmaster once and found it to be a stage for Taiwanese playing acting public speaking, with a dash of ESL practice thrown in. About as interesting as watching the laundry dry.

So far I've not found volunteer work calling for foreign language skills. I've approached the local ward office and found their attitude perfunctory at best.

Sports clubs, as the name implies, typically charge a fee, which is  unnecessary for a workout.

Take care meanwhile. By the way give I write short articles based on local news usually. If you'd like to be on my mailing list, then give me your email address.

When I lived in Taiwan I was studying on a one year scholarship at the Mandarin Training Center (MTC) at the National Taiwan Normal University, having just obtained a Post Grad Dipl in Modern Chinese in London. At the same time, when I first moved to Taiwan I was introduced to and lived with a wealthy family in the Songshan area. So overall I had a good start.

Initially, my circle of friends was pretty much other students from around the world at the MTC, and the family with whom I lived with. Of course I did explore much of Taiwan on my own and did meet both local Taiwanese as well as foreign nationals who were living and working in Taiwan.

I also did some part time English teaching to earn extra money, at the Taiwan Political University and some trading companies as well as individual students, so some of those people who I taught also became my friends. However, the majority of my friends were students at the MTC and the family that I was living with.

Outside of this social circle or friends, I did feel that as a foreign guy in his mid twenties, it was fairly easy to meet Taiwanese people. Those who had lived or studied overseas, usually in the USA, would be happy to become friends and have someone to speak English with. Those who didn't speak English would be happy to go out with me to see movies, see a live band, eat out or go to Danshui on the weekend.

I suppose that if I had worked in an office, probably I would have become friends with my colleagues. So I don't really see a problem in making friends. I believe it just depends how much effort you make. There are always ways but if you sit back and wait for the locals to come to you asking to be your friend then it probably isn't going to happen. So in that regard I would agree with Retired in Hualien's comments that it is much like making friends anywhere else in the world.