You can easily drive to Merzouga yourself or you can take an organized tour from Marrakech or Fez. Most of the tours include the transportation, food and 1 or 2 nights in the desert depending on how long you want to stay.
City of Marrakesh
Marrakesh is well-connected by public transport. Take a train from Casablanca or a bus from major cities; Marrakesh Menara Airport