Like Cara noted above, it really depends on your personal preferences. Would agree that you should look at Happy Valley. Have friends who live there and love it and its one of the few neighborhoods on HK island that we considered (with a baby and big dog). You get a local neighborhood feel, along with parks (and a nice indoor playroom which is a big deal when its hot and rainy), coffee shops, decent supermarkets, a bookstore, restaurants and other local businesses and you're close to Central/ Causeway Bay (insanely crowded but fun for shopping/ eating/ foot massages, etc.)/ an MTR link to anywhere in HK. There is a decent amount of green, although not much grass, except for the racetrack. Victoria Park in Causeway Bay is quite closeby and a wonderful green haven in the city with parks/ grass/ trees/ ponds. A great place for spending time outdoors with a baby on HK island.
I also know people who stay at Kowloon West, but to me the environment is so sterile and lacks character (but, this is just my personal opinion). Those who like in enjoy the convenience of living in quite new apartments with clubhouse facilities, atop a huge mall and MTR station with a big (expensive, even for us) western grocery store and lots of (pretty commercial) restaurants around. There is a "park" nearby with grass, but it also has a sterile, planned feel (again, my personal opinion). But it is a good size and nice for kids to run around.
You might also want to take a look at Stanley. Your budget is on the low side for the area, but you would be able to get into a nice 2 bedroom or small 3 bedroom at Stanford Villa (relatively new development) and a number of the other buildings near the waterfront and market. The market closes very early in the evening, so no noise issues. Stanley has a lot of expat families with young kids, a much lower population density than most areas on HK island, its very green and leafy, decent restaurants, a children's park and great indoor playroom (free), plenty of formal and informal playgroups for all ages, a good library, big grocery store (and a couple smaller ones), and plenty of room for children to play at several beaches (which are all within walking distance from anywhere in Stanley. Commute time to Central ranges from 20-40 minutes, with traffic playing a big part. It usually takes me about 35 minutes door to door in a taxi to get from Stanley to IFC/ Exchange Square in Central. Getting a taxi at anytime of day isn't a problem. There are good bus and minibus links to Admiralty/ Central/ Causeway Bay/ Chai Wan/ Aberdeen. If you're the kind of person who likes to get to know some neighbors, Stanley is a nice place for that kind of thing with so many families with small kids about.