The Singapore MRT is one of my favorite rail systems. It is cheap, immaculately clean, and easy to understand. As pointed out above, there are 2 transfers when travelling from Changi to the GH. However, the transfers are painless "walk across the platform" type transfers. The trains are frequent enough that you'll never be waiting for more than a few minutes. The entire journey is about 50 minutes.
The Orchard station is located about a block away from the GH. If you arrive from the airport, make sure you exit via the Scotts Rd. exit which puts you closest to the GH.