The first leg of the journey starts with reaching the Dubai International Airport. Then you can either -
Take a bus from Dubai to Hatta. The bus takes three hours to reach Hatta which is time-consuming as compared to the option of having your own car. Since it is a public mode of transport it is relatively cheap too. This can cost you around AED 15.
Alternatively, you can hire your own car which takes the lesser time to say two hours. This can cost you a little more than AED 90 per day. To reach, take the Maliha Road and follow the E - 102 route.
Since Hatta is a relatively small region, transportation options within Hatta is fairly limited. Most people opt to rent their own vehicles and this is considered to be the most convenient mode of transport. Public transit buses do ply routes from Dubai to Hatta and within Hatta itself, but they tend to be slightly irregular with frequency. Taxis are also another mode of transport that people use to get around within Hatta.