There are no no no official transfer/shuttle services from Nairobi so only options are to drive yourself, go on an organised tour or hire a taxi to/from the park.
Although Amboseli can be visited on a day trip from Nairobi, its not at all recommended since its approximately a 4 hour drive to get there. Organised tours for example 2 days / 1 nights are easy enough to find but pretty expensive.
I opted for a private transfer and made a deal with a tour company for approx usd 250. Pick up from hotel in Nairobi drop off at hotel just outside the park and return to Nairobi a couple of days later. I stayed at http://aalodges.com/aa-lodges/aa-amboseli/ at a very reasonable price – great place with a lovely pool and great views to Kilimanjaro. The park is best known for its large herds of elephants and a huge variety of birds. The park as such is quite flat and doesnt have much vegetation making it fairly easy to see the wildlife. Every afternoon the elephants will head towards the swamps and its quite a sight watching them feed and cool down.