I did this last summer, basically you take the MRT to Ang Mo Kio station then take bus 138. Painless and cheap but takes awhile as the bus makes lots of stops with the zoo being last.

The web site says it takes 30-40 minutes but I think it took closer to an hour, I remember a fairly long wait for the bus. There's a lot of restaurants and shops around the station so next time I plan on getting there early and checking out the area.

website's directions