You can pay most bills at any convenience store. Just hand it to the clerk and they will scan it and charge you the balance.
Paying rent and some other things is most easily accomplished by ATM bank transfer. You need the recipient's bank account number and the bank's ID code. With these in hand you can transfer money to your landlord or anyone else in Taiwan from most ATMs. This is also one way you can easily move money from your post office account to another account in Taiwan.