A very Singaporean shopping experience in the local markets. Have a Tiger and some noodles from the local eateries while you're there - very worhwhile.
Don't forget to visit Raffles for a Singapore Sling; I know it's very touristy but you'll regret it if you don't do it. The food is excellent as well.
Nowhere is too far to walk in Singapore because of the experience, just make sure you have plenty of bottled water with you. The new (2005) shuttle bus service around the city is fantastic, buy a "hop on-hop off" ticket; they stop near all the major sites, then you can walk some more.
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