I doubt that you will get a taxi much cheaper. There is the option of using the large shared taxis which is much cheaper. This is based on 6 people sharing, you can buy extra seats to make the seating arrangements far more comfortable. If you are a newcomer to Morocco, do be aware that these vehicles drive at break neck speeds, normally do not stop on route, it is one city to another. As there are three of you, you could purchase all seats, making it far more comfortable for all of you.
The second option is to use the train, you can view the fares and times on www.oncf.ma.
If you decide to use the Grande shared taxi, ask either your Hotel or Riad for information as to where the pick up point is from Fes.
I personally would use the option of a driver with your own designated stops.
Hope that you arrive safely,