Touch Move character

No problem

For that you will need to shoot a Raycast from the Camera to the $$anonymous$$ouse Position you can do that with:

 Ray ray = Camera.main.ScreenPointToRay (Input.mousePosition);
 RaycastHit hit;
     if (Physics.Raycast (ray, out hit, 2F)) {
                   //Then do some code to the hit
         }

after that you check the hitted object

     Ray ray = Camera.main.ScreenPointToRay (Input.mousePosition);
     RaycastHit hit;
         if (Physics.Raycast (ray, out hit, 2F)) {
                       if(hit.collider.gameObject....) //Here you just check if this square is ok for the player to go dont know how you are doing this system so i cant help you there sorry
             }

After Checking we go to the player moving

You will need to add the Nav$$anonymous$$eshAgent componentToIt

then you have in the script at the top to import

 import UnityEngine.AI;

then you make the player move to the square it clicked

         Ray ray = Camera.main.ScreenPointToRay (Input.mousePosition);
         RaycastHit hit;
             if (Physics.Raycast (ray, out hit, 2F)) {
                           if(hit.collider.gameObject....){
                                      GetComponent<nav$$anonymous$$eshAgent>().SetDestination(//Here the position of the square);
                           }
                 }

For making it easier you can also set the variable on void Start()

 public Nav$$anonymous$$eshAgent nav$$anonymous$$eshAgent;
 
 void Start(){
        nav$$anonymous$$eshAgent = GetComponent<Nav$$anonymous$$eshAgent>();
 }