Hi Maryann,
Chef Mickey's character dining is located right inside the Contemporary Resort and is a fantastic way to kick off a Disney vacation.
If you want to head to Downtown Disney from the Contemporary, the most direct route using on-site transportation is to ride a bus. These will be located right outside of the Contemporary and they'll drop you off at the entrance to Downtown. The water taxi's that you are referring to run from Saratoga Springs Resort back and forth to DTD, and also from Port Orleans Riverside to DTD. There aren't any that run from the Contemporary to there. However, you
can ride a ferry over to Fort Wilderness from your Resort if you think your son would enjoy the boat ride. A roundabout option for you, if you really would like to ride a water taxi to Downtown on your first night, would be to ride the monorail (or walk) to the Magic Kingdom, hop on a bus to Port Orleans Riverside, then ride the boat from there. You'll be adding about an hour to your trip though, so I don't recommend it.
Hope this helps and have a wonderful trip,
Kaylene