Hello Bianca! I'm a Singaporean with two adopted shelter dogs and I'd like to say that dogs are somewhat accepted here in Singapore.
We have dedicated dog runs (where your dogs can go off-leash in an enclosed space outdoors), and are certainly allowed on beaches here (leashed, unfortunately). As for renting apartments that accept dogs, that might be slightly harder to find as most landlords are quite picky, but with a good housing agent, he/she will be able to narrow down the options for you. It's not impossible! Just factor a little more time into your search.
As for bringing your dogs to the city (or suburbs), you'll have no issues there as long as they're leashed. Pets are not allowed on public transportation (bus/trains) but you can arrange for local taxis (Uber/Grab) for this - just make sure you've confirmed your situation with the driver.
Do let us know if you have additional questions! Welcome to Singapore in advance by the way!