How do I transition among multiple animations?

Mecanim is kind of created exactly for this type of set up and others. I'd strongly suggest getting into it and learning how to control the transitions with code. However if you'd still like to keep using your script - you'll have to show/explain how it is currently set up so others can provide input on how to proceed.

Thank you for responding. I'm having trouble figuring out how to do things like allowing my "alert" animation to keep being transitioned to if my player has attacked in the past 5 seconds. I have been able to make my character shoot his bow then play the alert animation, but if I walk during this animation and stand still I go back to idle when I'd like to go back to alert for 5 seconds after the character shoots his bow. I'm also having trouble with allowing my player to shoot his bow again once he is in the alert animation.

Here is my current script:

 using System.Collections;
 using System.Collections.Generic;
 using UnityEngine;
 
 public class player_1_controller : MonoBehaviour {
 
     //movement variables
     public float maxSpeed;
 
     //jumping variables
     bool grounded = false;
     float groundCheckRadius = 0.2f;
     public LayerMask groundLayer;
     public Transform groundCheck;
     public float jumpHeight;
 
     Rigidbody2D myRB; 
     Animator myAnim;
     bool facingRight;
 
     //arrow shooting variables
     bool firing = true;
     public float arrowShoot;
     public Transform arrowTip;
     public GameObject arrow;
     public float fireRate = 0.5f;
     public float nextFire = 0f;
 
     //alert variables
 
     // Use this for initialization
     void Start () {
         
         myRB = GetComponent<Rigidbody2D> ();
         myAnim = GetComponent<Animator> ();
         facingRight = true;
     }
     
     // Update is called once per frame
     void Update () {
         
         if (grounded && Input.GetButtonDown ("Jump")) {
             grounded = false;
             myAnim.SetBool ("isGrounded", grounded);
             myRB.AddForce (new Vector2 (0, jumpHeight));
         }
 
         //player shooting
         if (grounded && Input.GetButtonDown ("Fire1")) {
             myAnim.SetBool ("arrowShoot", firing);
             Invoke ("fireArrow", 1);
             }
     }
 
         void FixedUpdate() {
         
         //check if we are grounded - if not, then we are falling
         grounded = Physics2D.OverlapCircle(groundCheck.position, groundCheckRadius, groundLayer);
         myAnim.SetBool ("isGrounded", grounded);
 
         myAnim.SetFloat ("verticalSpeed", myRB.velocity.y);
 
         float move = Input.GetAxis ("Horizontal");
         myAnim.SetFloat ("speed", Mathf.Abs (move));
 
         myRB.velocity = new Vector2 (move * maxSpeed, myRB.velocity.y);
 
         if (move > 0 && !facingRight) {
             flip ();
         } else if (move < 0 && facingRight) {
             flip ();
         }
 
     }
             
         void flip () {
         facingRight = !facingRight;
         Vector3 theScale = transform.localScale;
         theScale.x *= -1;
         transform.localScale = theScale;
 
     }
 
     void fireArrow() {
 
         if (Time.time > nextFire) {
             nextFire = Time.time + fireRate;
             if (facingRight) {
                 Instantiate (arrow, arrowTip.position, Quaternion.Euler (new Vector3 (0, 0, 0)));
             } else if (!facingRight) {
                 Instantiate (arrow, arrowTip.position, Quaternion.Euler (new Vector3 (0, 0, 180)));
             }
         }
         playerAlert ();
   }
 
     void playerAlert () {
 
         firing = false;
         myAnim.SetBool ("arrowShoot", firing);
     }
 }