How can I get my character to move and strafe relative to the way its pointing?

I originally used transform.Translate to handle movement but since that didn't account for collision I instead switched over to controller.Move. That does account for collisions but does not account for the direction the character is facing. using UnityEngine; using System.Collections;

 public class PlayerClass : MonoBehaviour
 {
     private float hspeed = 10f;
     private float vspeed = 0F;
     private float gravity = 1F;
     private float test = 0;
     private Vector3 moveDirection = Vector3.zero;
     private float hdirection;
     public bool isaccending=true;
     private float movspeed=0;
 
     // Use this for initialization
     public float movementSpeed = 10;
     public float turningSpeed = 60;
     void Start(){
         
     
     }
     void Update() {
         CharacterController controller = GetComponent<CharacterController>();
         float horizontal = Input.GetAxis("Horizontal") * turningSpeed * Time.deltaTime;
         transform.Rotate(0, horizontal, 0);
         float vertical = Input.GetAxis("Vertical") * movementSpeed * Time.deltaTime;
         float strafe = Input.GetAxis("Strafe")* movementSpeed * Time.deltaTime;
         Vector3 movement= new Vector3(-vertical, 0, strafe);
         
         controller.Move(movement);
         isaccending=controller.isGrounded;
         if(controller.isGrounded){
             vspeed =0;
             
         }else{
             vspeed-=gravity* Time.deltaTime;
         }
         
         controller.Move(new Vector3(0,vspeed,0));
         
         
     }
     
 }