How do I create the animator and the character controller for this situation?

Okay I'll try to give you some guidance here. First, you'd setup the Animation controller kind of like this:

Each of the transitions to the directional states has a condition set so that it only moves when the relevant bool is set and shooting is false. The transition to shoot happens whenever shoot is true.

Then, I'd have two separate scripts. The first script will set/unset movement variables as needed, and the second will read the input. Here they are:

 public class AnimController : MonoBehaviour{
     private string m_LastDirection;
     private Animator m_Anim;
     
     private void Start(){
         m_Anim = GetComponent<Animator>();
     }
     
     public void SetDirection(string dir){
         //Dont do anything if the animation is already set for that direction
         if (dir != m_LastDirection){
             //Set all movement bools to false
             m_Anim.SetBool("left", false);
             m_Anim.SetBool("right", false);
             m_Anim.SetBool("up", false);
             m_Anim.SetBool("down", false);
             
             //Move in the given direction
             m_Anim.SetBool(dir, true);
             
             //Save which direction we were last set to move in
             m_LastDirection = dir;
         }
     }
     
     public void SetShoot(bool val){
         m_Anim.SetBool("shoot", val);
     }
 }
 
 public class Character : MonoBehaviour{
     private AnimController m_AnimController;
     private float m_ShootingTime; //This is the time the shooting animation takes
     private void Start(){
         m_AnimController = GetComponent<AnimController>();
     }
     
     private void Update(){
         if (Input.GetKeyDown(KeyCode.Left)){
             m_AnimController.SetDirection("left");
         }
         if (Input.GetKeyDown(KeyCode.Up)){
             m_AnimController.SetDirection("up");
         }
         if (Input.GetKeyDown(KeyCode.Right)){
             m_AnimController.SetDirection("right");
         }
         if (Input.GetKeyDown(KeyCode.Down)){
             m_AnimController.SetDirection("down");
         }
         if (Input.GetKeyDown(KeyCode.Space)){
             m_AnimController.SetShoot(true);
             StartCoroutine(WaitForShoot())
         }
     }
     
     private IEnumerator WaitForShoot(){
         yield return new WaitForSeconds(m_ShootingTime);
         m_AnimController.SetShoot(false);
     }
 }

Note that this code is not tested, but it should help get you started.