You can enter Singapore without any problem. Just head towards the arrival hall after disembarking. Fill out an arrival card (will be distributed onboard), head to immigration and you are landside in no time. Thats not considered a stopover, just a long layover.
In PEK you should ensure that your luggage is checked through. If not you gat to collect it in SIN, but you would be able to check it in at the early check-in counter in SIN.
You can take the MRT or Cab to town. Cab would be easier, its efficient and cheap, you dont need to worry to get ripped off.
I would go to the Esplanade to get a first impression and see the skyline at MarinaBay. Later you can have a stroll on Orchard Road to see the glittering malls. After there would be time to head to Chinatown look around and eat diner there before you return to the airport.