Forget the bus - it was a nice idea at the time.
The government's tourist information page lists only cars, the railway, and bicycles as modes of transport for Hualien.
They probably know something we don't: tour-hualien.hl.gov.tw/Portal/Co ... ea=3&id=67
I have been using bicycles and the occasional taxi for inner-city transportation and the railway for long-distance travel.
Long-distance buses are available, too - i've seen them, know some of their routes, and know where the terminal is.
But i have no recollection of public buses or bus stops. :idunno:
( EDIT: Looks like i got confused here and thought you were talking about public transport within Hualien City - if you go to the bus ticket office, near the railway station, you should be able to get all the info you need about the long-distance buses, including those going to Ilan and Taipei. )