Some ideas:
Shopping:
- Orchard Road; the national shopping belt. Branded boutiques galore.
- City Hall area; The Adelphi is a shopping mall full of audio retailers, just across the road is Funan for computers and electronics, just across the road again is Peninsula for camera equipment, both new and second hand.
Cultural/historic:
- City Hall area; some historic buildings since the British Colonial days are still around, the contrast between 19th century and modern architecture can be interesting. There are a number of museums in the area too like the Asian Civilisation Museum and the National Museum.
- Changi Museum; near the airport, where one can learn about how life during WWII Japanese occupation was like for civilians and the Allied POWs.
- Chinatown; pre-war buildings and lots of chinese shops and eateries.
- Tiong Bahru; one of the oldest estates, has become a mix of old and hipsterish culture.
Family entertainment:
- Sentosa; an island resort with many attractions. Universal Studios is located here. Can be a bit pricey though.
- Singapore Zoo and Night Safari; open concept zoo that's highly rated.
- Singapore Flyer; giant ferris wheel, great view of the waterfront.
Eating:
- Maxwell Road food center, lots of local staples here.
- Tiong Bahru, the food center there has some nice chinese food.
- Clarke Quay/Boat Quay; more for clubbing/drinking.
- Katong; famous Katong Laksa.
- East Cost Park; many seafood eateries there like Jumbo and Long Beach. Must try the chili crab.
Nature:
- Mount Faber; have a good view of Sentosa from the top, can take a cable car from here to Harbourfront, then to Sentosa and vice-versa.
- Sungei Buloh; marshland that's home to a wide variety of wildlife, especially birds.
- Pulau Ubin; a rustic island off Changi, accessible by ferry, great for cycling, seafood and wildlife spotting at Chek Jawa.