Some help with a movement code

I am making a variant of flappy bird.
The idea is to have my character to go up and down with mouse click, like Flappy Bird. It would start in the middle, without moving.
Once I press the button the player will be launched up. When it reaches the top of its way he goes down.
A couple of problems I had are.
Sometimes the player will just go up all the time.
Sometimes it will just go down.
I do not want for it to go up and because of the gravity, go each time less high and after go down, when the speed of it going up ends. I would like to have a movement like:
Mouse click - Go a bit up, then go down with the speed increasing as it goes down. Something really aggressive, lets say

Script:
using UnityEngine;
using System.Collections;

public class PlayerMovement : MonoBehaviour {

	private bool hasStarted = false;
	private bool goingUp = false;
	private bool goingDown = false;
	
	// Use this for initialization
	void Start () {
	}
	
	// Update is called once per frame
	void Update () {
		if (hasStarted == false) {

			this.transform.position = new Vector3(1,6,-1);
			
			// Wait for a mouse press to launch.
			if (Input.GetMouseButtonDown(0)) {
				hasStarted = true;
				goingUp = true;
			}
		}
		if (hasStarted == true) {

			if (Input.GetMouseButtonDown (0)) {
				goingUp = true;
			}

			if (goingUp == true) {
				print ("Launch Bird");
				rigidbody2D.velocity = new Vector3 (0, 5, 0);
				goingUp = false;
			}
			if (goingDown == true){
				rigidbody2D.velocity = new Vector3 (0, -10, 0);
			}

			if (goingUp == false) {
				goingDown = true;
			}
			if (goingUp = true) {
				goingDown = false;
			}

		}


	}
}

In this case I think you should be really using the physics engine.

First, add a rigidbody to the bird. That will make it use the physics engine, and fall reallistically, thanks to the physics engine.

Now that you have a rigidbody you should add force to it instead of “translating” it, for more realistic movement like the one you want (Starts moving fast, slows done gently, when it stops it starts to fall slowly then faster).

IMPORTANT: Use FixedUpdate() when working with physics!!

Now a simple code for adding force in the global up axis when mouse is pressed:

//Force variable
public float force = 500.0f;

void FixedUpdate() {

   //If mouse button is pressed down...   
   if(Input.GetKeyDown (KeyCode.Mouse0)) {
      
      //Add force upwards in the global axis
      rigidbody.AddForce(Vector3.up * force, ForceMode.Impulse);
   }

(ForceMode.Impulse) adds an instant force. ForceMode.Acceleration, the default one, would accelerate slowly upwards.

As you can see in my code, using the physics engine when appropiate will make things easier especially when working with falls, slow accelerations, etc.

Hope I helped!

Maybe this works for you:

	float currentTime;
	float fallTime = 0.5f;
	float moveSpeed = 1f;
	Vector3 movement;

	void FixedUpdate(){
		if (Input.GetKeyDown (KeyCode.Mouse0)) {
			movement.y = moveSpeed;
			currentTime = Time.time + fallTime;
		}
		if (Time.time > currentTime)
			movement.y = -moveSpeed;
		rigidbody.velocity = movement;
	}