There are tons of non-touristy things to do in Singapore but as I said, you need to know where to look.
For example, if you look at Google Maps you might notice a huge green dot between Chinatown and Orchard. That is Fort Canning Park- one of my other favorite nature spots in Singapore. As the name suggests, it's a natural park managed by NParks (short for National Parks Board, a governmental body that strives to not only preserve Singapore’s green areas but also educate people about appreciating nature).
The trails at Fort Canning can be quite steep, but they’re less challenging than the ones in Bukit Timah Nature Reserve. Located in the city centre, the park is much smaller than other national parks, but since it’s in the heart of the city you can easily visit Boat Quay and the other cool things to do in Singapore.
In the neighborhood is also an area that was once used as a British military barracks, which is now home to music events and is a food lover’s paradise.
The National Museum and the Singapore Philatelic Museum are also nearby and worth visiting.