There is a difference between the destruction staff item and the Destruction Staff skill line.
The fireball / 3x fireball is the basic attack of the staff itself, much like a sword will cut / power cut or a bow will shoot / power shoot. You will get that ability without investing anything into the skill associated with the item.
However, as you use items in ESO you will develop an associated skill. For a greatsword, you will learn 2-Handed Weapon; for a bow you will learn Archery and for a destruction staff you will learn Destruction Staff. By learning this skill you will make certain abilities (5 actives and 5 passives I believe) available for you to purchase with skill points. The active abilities are similar to 'spells' as you know them from other games, and will likely involve more spectacular effects. The passive abilities will improve your destruction staff's attacks and spells.
Also, for what it's worth, you do not morph your basic attack. You morph abilities. That is, the 5 active abilities you can gain from the skill line.