Take the MRT to Bras Basah Station and exit at the Singapore Management University grounds.
There you have a good view and can visit the National Museum of Singapore and SOTA School of the Arts (A junior and senior high school).
You can then walk to the city hall area (head East) where the buildings and churches along the way are from the colonial era and have beautiful architecture with tall white columns. Visit CHIJMES (a convent school turned dining area) and St Andrews Church.
Head south and you can see the Padang (a large green field where on sundays there may be cricket matches) and the Parliament House (Under renovations right now)
You should also head towards the Singapore river where you can see the Old Parliament House and the Asian Civilisations museum (a well curated museum showcasing asian culture and history)
Here you can enjoy the bay from 3 vantage points:
One. One Fullerton - Its a lovely place for dining and you can vist our national icon the Merlion. Take the underpass from Fullerton hotel (A converted colonial Post Office) to admire the grand spacious architecture inside the building.
Two. From the Esplanade (Concert hall and theatre) - There are free performances in the evenings (from about 6.30pm onwards, one indoor one outdoor) and you should visit the library as well as the roof for a nice view of the bay.
Three. From the Marina bay Sands - There is a spectacular light show at 8pm and 9.30 (highly recommended) and you can try to get to the top floor and have a fantastic view of Singapore and the infinity pool. There is also the Art Science Museum.
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Visit the National Library (a tall building with plants all over it) between bras basah and Bugis MRT, and stop by at Bugis Junction (An airconditioned Street) before crossing the street to Bugis Village (A market where you can also get souvenirs) and the bugis area itself where there is an Hindu and buddhist/taoist temple side by side (Some people from each religion cross over to offer prayers to each others' Dieties)
Stop by LASALLE college of the arts before heading to the Little India District.
Head east from little india to where there is massive construction. Hidden inside is the Thieves Market (like a bazaar selling old possessions and relics many thought to be stolen wares or trash picked)
Then continue to Arab street (Arab and SE-asian textiles, perfumes and art) and visit the Huge Gold domed Sultan Mosque. There is also a small lane called Haji lane where the local free spirited subculture is found.
Beautiful shopping complexes to visit include
Orchard Central - Ride long escalators outside the building to the roof top at night for a magical experience and exquisite garden and dining areas.
Area outside Ion Orchard
Visit the botanical gardens - a really nice place for a picnic or a stroll in the evenings and Dempsey Village - Colonial bungalows and homes turned into lifestyle stores and a really fancy and beautiful place for dinner. The US embassy is there too. =)
Most visitors spend all their time at tourist places and many CS-ers live in areas where there are many expats. For a true singapore experience, visit the Towns (where most singaporeans live and where public housing and amenities for our people are found. I highly recommend visiting:
Toa Payoh Town, singapore's second oldest developed town and walk the market streets and shops. If you can, visit the tallest public housing block (just opposite the library) and visit the sky gardens. (land scarce singapore has led to development of open spaces in the air) This is the second of the experimental development of public housing (the first is Duxton in Chinatown)
Punggol Town: Take the MRT to Punggol station and transfer over to the LRT. This is the newest town that is in the process of completion and features the culmination of the latest urban development and technologies. (I always find this area like a Sci-fi story)
Singapore is only 45 km across, but about a quarter of our land are nature reserves and if you look on google maps almost half of our island is green. It's nice to visit some of the nature parks and walks to get faraway from the busy urban landscape.
Pulau Ubin - A rustic island that still preserves our last farms and village style homes. People take a bumboat from Changi Jetty Point and cycle around the island.
Sungei Bulloh Mangrove - A mangrove reserve, that as a biologist, i feel is important for people to learn and understand. These specially adapted forests exist at the edge of lands, where salt and fresh water meet. Its often muddy and full of bugs and many land developments around the world clear (because its deemed ugly) without realising that it is teeming with huge biodiversity and protects the land from encroaching ocean waves (and tsunamis)
Southern Ridges Walk - A chain of forested hills, starting with Mt Faber (where spectacular night views and cable cars to sentosa are located) and linked by pathways where there is a beautiful and high Helix bridge, tree top walks and Hort park (a horticulture park featuring testbed/ ideas for tropical landscaped gardens)
Gardens by the Bay and Marina Barrage. GbtB is a billion dollar garden that is nearing completion. There is a little known view of the Singapore skyline from the marina barrage that is simply splendid. Personally I've never seen a city quite like this (perhaps maybe Isfahan in Iran) where mankind's urban development is balanced with man-tamed ecology. Good especially during the evenings (about about 7pm when the lights switch on).
Singapore has no natural resources, and our economy is built purely on people power. You can take a visit to the west of our island to Buona Vista or One North MRT where you can find our knowledge industries.
Biopolis and One North - This is a governmental initiative to jump start a knowledge-based economy and you can see how the country creates an entire economy from nothing, into mega complexes housing people and research facilities.
National University of Singapore - you can visit our highest ranked university and see our take on education. It's huge and behaves like a city itself with housing, internal shuttle buses and a University Town.
Science Center - Its really old and dilapidated, but the iMax theatre is worth visiting if your son is keen on catching short 20min documentaries on a huge dome screen. (I personally love the experience, better than 3D but I'm sure US has much better technology on this)
14 Ulu Places to Visit