Character is running through terrain instead of over it

I've been having trouble getting my player character to run over terrain. I've tried a few different methods including transform.Translate, rb.Velocity, and rb.MovePosition. Velocity and MovePosition both give me a problem of the character speeding across and also floating above the terrain and ground. Translate does not give me these problems, but will not allow the character to run up terrain and only works properly when gravity is turned off and IsKinetic is turned on.

Here is the script I am using currently - please do let me know if any of you see a way to modify this to allow the character to run over terrain:

 public float speed; 
 private Rigidbody rb; 

 void Start () {
     rb = GetComponent<Rigidbody> (); 
 }

 void FixedUpdate() {
     moveMonkey (); 
 }

 void moveMonkey()
 {
     float moveHorizontal = Input.GetAxis ("Horizontal"); 
     float moveVertical = Input.GetAxis ("Vertical"); 

     Vector3 movement = new Vector3 (moveHorizontal, 0.0f, moveVertical); 
     transform.rotation = Quaternion.LookRotation (movement); 

     transform.Translate (movement * speed * Time.deltaTime, Space.World); 
     }