How easy/hard is it to backpack to Taiwan knowing only English?

I rode my bicycle around the whole of Taiwan. English is spoken a bit more in Taipei than other places, but it's hit-or-miss everywhere. I found a tea shop in Taipei where the staff did not understand "tea" or any other relevant words...but a helpful local next in line stepped up to translate. As with many places in the world, younger people (under 30) tend to speak English better, and are also more interested in doing so (for their own practice).

If you learn to pronounce "xie xie" (thank you) you will not be received badly. For a short trip, cramming other phrases may not be particularly worthwhile--just do as you would do anywhere with a language barrier and gesticulate.

If you venture outside Taipei, you may encounter some aboriginal Taiwan people. These people are not Chinese, and if they speak to you it is likely to be in English. Business people in any city will also speak English to you, often without prompting.

If you do try to learn some Chinese, you should realize that knowing how to pronounce things doesn't help you to read them. It is occasionally difficult to find a hotel simply because you cannot read the signs, or an international brand name store because Google Maps only has it in Chinese (e.g. Carrefour in Taipei is listed on Google Maps using a Chinese transliteration and not the original French name which is the largest sign on the actual store).

You should enable Chinese keyboard entry on your smartphone using Pinyin. At least this way you may sometimes be able to type things from guidebooks or get a local to type something in to translate for you. I wish I had done this on day one instead of day ten.

Don't let any of this deter you from visiting Taiwan...but if you want something just one notch easier on the difficulty scale, you can try Korea. :)

I've travelled around rural parts of Russia, Mongolia, China and other countries where English is not widely spoken and few tourists go without knowing the native languages myself. As others have said, it's surprising how much you can communicate without using language when you have the need.

My advice is to bring along some phrase books which list phrases in both English and Chinese; what I have done before now is found the phrase I want to use in English and shown others the translation. I've even held conversations with others this way without either of us knowing the other language :)