The paddle steamer or boats (see pictures) are starting at the Market Square next to the ferry to the fortress. It is absolutely the best option to go to the zoo. A nice short sea trip through the harbour.
There are other boats starting a mile further north (Hakaniemi) so be sure you start your return trip from the zoo from the same pier that you arrived to.
Bus 11 starts from the east side of the Railroad Station twice a hour and takes you around the bay to the zoo gate/bridge on the north side. As it is midsummer there is risk the bus might be quite full. I prefer the boat trip.
Katajanokka is #5 on the map. Easy straightforward walk to the Market Square.
Melkki (aka LMA) is #4 on the map. There worst one in this case. Shuttle bus to downtown plus walk along Esplanade Park to Market Square is the easiest option.
Hernesaari is #1 & #2 on the map. Bus #16 will take you to Market Square. In fact if you are adventurous and confident in map reading , bus 16 would, after a rather long trip, take you all the way to Kulosaari island (stop: "Marsalkantie") from where its about a mile walk cross another island to the zoo.
Hope it helps.