1. Queenstown (the library, as well as Alexandra Village)
2. Little India (Indian temples)
3. Toa Payoh (old HDB flats)
4. Tiong Bahru (Tan Tock Seng's grave, etc)
5. Kampong Glam (Malay history)
All these are good suggestions. I just have one personal favourite I'd like to add on (a schoolbus or car is advisable): a road trip tracing the battle of Singapore, from the arrival of Japanese soldiers to the surrender of the British.
Start from the Lim Chu Kang jetty, close to where the Japanese first made their inroad into Singapore.
Source: Tails from the Lion City
Then, pass by Tengah airbase and go to the southern ridges, where you can see some of the old cannons built along the shore. Contrary to popular beliefs, these cannons could be (and were) turned around during the Japanese invasion, but were generally ineffective.
Climb Bukit Chandu. It was around here that the Battle of Pasir Panjang was fought. Lieutenant Adnan Saidi bravely led a team of 42 people in resisting the Japanese onslaught, even though his platoon was greatly outnumbered and outgunned. Take a look at Reflections at Bukit Chandu, a museum contained within an old colonial era bungalow, near the top of the hill itself. The museum is open from Tuesdays to Sundays.
Source: NHB website
Then make your way to Bukit Timah, where the British valiantly made a counterattack on 14th February 1942. End in the Ford Motor Factory, where the Japanese surrendered.
Source: Memories at Old Ford Factory website
By the way, the table on which British General Percival surrendered to Japanese General Yamashita is currently in the Australian War Memorial (Canberra, Australia). The Singapore government has wanted to bring it back to Singapore, but naturally, the Australians prefer to keep it in Australia.
Edit: CNN has an interesting suggestion. Start from British colonial mansions in Sembawang, then travel through the green heart of Singapore (e.g. MacRitchie Reservoir Park, where you can see Lim Bo Seng's resting place, among other things), then end in the south (similar to what I suggested). The writer did the walk through Singapore's north-south axis on foot, but it is probably better to use some form of motor transportation.
