Character bounces off walls, and drops to the ground on movement

Hi guys, I am making a click to move game. Think of Diablo. At the moment the character only moves with the click of the mousebutton, and has a rigidbody attached to it to apply some physics. The problem is that whenever the character moves at a faster speed it falls (Not through the ground), but falls to ground, or whenever it collides with a fall it will bounce and fall. I just want it to collide and stop.

Right now I am moving my character with transform, but I would also like your input if I should move it with Rigidbody instead. In the end it is going to be a mobile game where you move the character with your finger.

Just in case, after I have managed to fix this I will apply animations to the character.

Here is a little gif showing what is happening: https://gyazo.com/53f443146b859b56011e939f162c5bf6

I can imagine that I can set the speed to 0 when my object is colliding with 1 particular object, but do I really have to do that FOR EVERY OBJECT? It also doesn't fix the problem that my character walks and falls.

Here is the movement script I found off the internet:

 using UnityEngine;
 using System.Collections;
 
 public class ClickToMove : MonoBehaviour {
     private Transform myTransform;              // this transform
     private Vector3 destinationPosition;        // The destination Point
     private float destinationDistance;          // The distance between myTransform and destinationPosition
 
     public float moveSpeed;                     // The Speed the character will move
 
 
 
     void Start()
     {
         myTransform = transform;                            // sets myTransform to this GameObject.transform
         destinationPosition = myTransform.position;         // prevents myTransform reset
     }
 
     void Update()
     {
 
         // keep track of the distance between this gameObject and destinationPosition
         destinationDistance = Vector3.Distance(destinationPosition, myTransform.position);
 
         if (destinationDistance < .5f)
         {       // To prevent shakin behavior when near destination
             moveSpeed = 0;
         }
         else if (destinationDistance > .5f)
         {           // To Reset Speed to default
             moveSpeed = 10;
         }
 
 
         // Moves the Player if the Left Mouse Button was clicked
         if (Input.GetMouseButtonDown(0) && GUIUtility.hotControl == 0)
         {
 
             Plane playerPlane = new Plane(Vector3.up, myTransform.position);
             Ray ray = Camera.main.ScreenPointToRay(Input.mousePosition);
             float hitdist = 0.0f;
 
             if (playerPlane.Raycast(ray, out hitdist))
             {
                 Vector3 targetPoint = ray.GetPoint(hitdist);
                 destinationPosition = ray.GetPoint(hitdist);
                 Quaternion targetRotation = Quaternion.LookRotation(targetPoint - transform.position);
                 myTransform.rotation = targetRotation;
             }
         }
 
         // To prevent code from running if not needed
         if (destinationDistance > .5f)
         {
             myTransform.position = Vector3.MoveTowards(myTransform.position, destinationPosition, moveSpeed * Time.deltaTime);
         }
     }
 
 
 }