Hey,
Welcome to Singapore and hope you enjoy your stay in here.
Sports, it depends if you're a indoor or outdoor person. Sentosa, West/East Coast Park, MacRitche/Seletar et cetera reservoir have good running areas and sports for you to do, like sailing, wake boarding, man-made water surfing, cycling, sun, sand and sea.
Nature, Bukit Timah Hill is a good relaxation spot for your inner soul'
Hope you've fun!