It depends a bit on which part of Damansara you'll be living in, but for the most part, it is served by buses that run roughly every 30-45 minutes during the day (and none after midnight). In reality, if you need a bus after about 9 pm, you'll have a long wait. Taxis are available and you'll need to use them - insist that they use their meter or it can be hard on your budget. From parts of KL, it is difficult to get a taxi that will use a meter (like KLCC or the Pavillion), in which case, you'd be ahead by taking the train to the closest station to home (probably Bangsar) and then taking a taxi from there.

Personally, if I was to move to Damansara, I'd get the car. If you are buying a place there and coming on the MM2H visa, you're allowed to import a car tax-free or purchase one tax-free once you're there. Might be something to consider if this is going to be a permanent move for you.