We live in Singapore and moved from HK last September. Singapore was also our first trip with our bub when he was 9 months.
There is nothing to worry about when here. The medical care is actually better than HK's and anything you would find in HK, you can find here. It's a fast trip and a very stroller friendly city.
Fun things to do - Sentosa Island (beaches, aquarium, Resort's World just opened and soon after Universal will), Singapore Zoo is great for kids, the Botanical Garden and its Children's Garden.
It is hotter here than in HK, but less humid. Cabs are very easy to get and you can ask any restaurant/hotel to call one for you and the cab will come in 5 minutes.
Singapore is clean, safe and easy to get around. It's the perfect trip for you to build confidence in traveling with your little one. Since then, we did Bangkok, Vietnam and Phuket with our son who is now just 2!