Places I personally started off with were the Singapore River, Esplanade & Merlion, Chinatown...shoot at these places in bright daylight first to get used to your equipment, decide what aperture value is best for your general images, and to find your own way of handholding the camera steadily and comfortably.
When you've gained more confidence, take out a tripod to the same places, and even Clarke Quay, and try long exposures. If you hate using the tripod, bump the ISO up to 800 or 1600, switch the camera to Program mode (or equivalent) and shoot night photographs handheld.
Moving further, if you find yourself with a telephoto zoom, go to places like the zoo. Personally, I've had great difficulty shooting all these moving subjects, especially under lighting that's usually quite bad, and at long focal lengths. Focus gets tricky, and handholding is naturally more challenging at lower shutter speeds.
That said, there's no fixed way to learn photography because everyone does it differently. We can only recommend, then it's up to you to decide what you want to attempt first.
The bottom line from most people seems to be: shoot more, and shoot lots. Try different conditions of light, different times of the day, different angles...go high, very high (aerial photos, if you're rich enough to hire a helicoptor), go low, shoot for repeating patterns, bright colors, lines, abstract etc. And cute stuff always works well...all the little animal shots.
