Goals give us something to strive toward and look forward to. People become inspired by a challenge. By creating a challenge or a goal around customer service you might be surprised just how well people respond, and the fulfillment they feel, when they have met the challenge and achieved the goal.
Regarding the challenge/goal, it must be realistic and attainable. Base it on something from the past. For example, a company may have an internal policy that all customer calls are responded to within 24 hours. A new goal or challenge would be that all calls before 3:00 are responded to within the same day. Is that reasonable? Talk to your employees about it. Get their input, and more importantly, their buy in. And watch how they rise to the occasion.