Obviously, you need to find the nearest 7-Eleven. It should be fairly easy as they are located on almost every block. Find the Ibon machine which looks like an ATM and don't get intimidated by the Chinese. As you may have guessed it has no English.
Purchase a Voucher
#1 Click the Dollar sign in "儲值/繳費" box, meaning "Stored value/Payment".
#2 Choose "電信加值" which means "Add value to telecommunication".
#3 Select "台湾大哥大" (Taiwan Mobile). You can choose a different mobile operator if you have one. The rest of the steps should be similar. Go figure.
#4 Click "儲值" (Stored value) again.
#5 Choose the second box "上網儲值" which means "Add value to the Internet".
#6 Pretty useless instruction screen if you don't speak Chinese.
#7 Choose the package you want to purchase and click "下一步" (Next). I always go for 1.2GB for NT$180 which is valid for 2 months and usually lasts me for around 30 days.
#8 Confirm your selection and proceed to the last step.
#9 Grab the receipt from the machine, bring it to a cashier and pay for it.
Top-up Internet Data
The cashier will give you a receipt with a voucher full of Chinese text. Don't bother to decipher it, just try to locate a 14 digit number in an upper area of the voucher.
Call a Toll Free number "867" and either follow instructions in English or press #2 and enter the 14 digit number.
Upon entering the number you can safely end the call. You will get an SMS saying that you need to restart your phone to use the credit.
Sometimes it's really needed to restart the phone before you can use the Internet again. If you have issues using it, just turn on the "Airplane mode" and then turn it off. It's usually good enough.
If you have any issues using the Ibon machine, just ask cashiers they will help you or do the whole process for you.