How to make object move forward and up simultaneously

So .velocity is a vector. It has components .x .y and .z to indicate movement in all directions in 3D space. What you are doing now is setting that vector to 2 different values one after the other and only the latter applies like you have noticed

 GetComponent<Rigidbody>().velocity = new Vector3(0, 0, 1)* speed;
 GetComponent<Rigidbody>().velocity = new Vector3(0, 1, 0) * speed;

What you want to do is set the value to be a vector that has the correct values for each component

 Vector3 v = (Vector3.up + Vector3.forward) * speed;
 GetComponent<Rigidbody>().velocity = v;
 
 // or 
 
 GetComponent<Rigidbody>().velocity = new Vector3(0, speed, speed);

write it any way you want as long as you set .y and .z to the value you want. Also not that the way you are doing it would result in the balloon traveling faster when it's going up or down since the length of a vector (0,speed,speed) is longer than that of (0,0,speed). To solve that you should normalize the vector (make its length 1) before multiplying it with speed

  Vector3 v = (Vector3.up + Vector3.forward).normalized * speed;
  GetComponent<Rigidbody>().velocity = v;