Unfortunately while the Birdpark is a bit closer in to town than the Zoo (and thus Night Safari), it's more difficult to get to by public transport as there's no direct route to it. Details are on the website ; the best way for most of us is to take the MRT (mass rapid transit – subway/trams) from somewhere on the right line and then transfer to a bus.
The MRT is no trouble to use – everything is at least bilingual (usually trilingual) and the system is straightforward. One small point of confusion is that you'll find yourself paying $1 SGD more than you thought the ticket price was. That's actually a deposit for the ticket card; when you're done using the MRTs you take it back to one of the ticket machines such as you bought it from, and it'll refund your $1.
Make sure you have a couple of dollars in coins for the buses; the locals use multi-trip passes exclusively and accordingly they don't have a lot of cash onboard, so usually only accept up to about $2.
Once at the Birdpark you have the option of taking the “panorail” hanging monorail tour. I took it and I would suggest you don't bother unless you're too tired to walk around; they announce what's at the stations and mention a couple of the other birds you're going past, but that's about it – there's no informative commentary. And the Birdpark is quite compact, so you can quite happily walk around it (and are likely too anyway if you want to see birds up close).
There's a good selection of birds, but the ones in cages won't excite (I'm not keen on animals in cages anyway); the best bits are the big open sections, in particular the flamingo lake and nearby pelicans (who may casually try eating your shoe) and swans, the big waterfall aviary, and the plethora of little colorful birds in the Lory Loft. The owls in the night-and-day-swapped World of Darkness were also a treat to me due to the lack of them in NZ.
Entry to the Birdpark costs $18 SGD, I'd skip the $4 panorail. Recommended if you like birds!
Catching the train and bus there was no problem. Catching the bus back didn't work out so well – I did what seemed like the logical thing and walked across the road and went along till I found a bus stop.
Frustratingly, although this was quite definitely on the route of the same bus I came in on – those buses were whizzing past regularly – they weren't listed on the stop, and they refused to stop at it. I walked along to the next couple, but it wasn't listed there either, and wasn't stopping.
Head scratching.. confusion.. annoyance.. acceptance.. gps, walking, gps, walking, walking, trams.
So don't ask me how to get back! I guess ask the people at the Birdpark (I was already some way back by the time I gave up, so didn't return to ask).