To clarify further, in order to cancel a spell or attack, the movement has to occur during the combat/cast steps. In the case of a spell, it would need to occur after step 1 cast spell and before step 3 resolve spell. In the case of an attack, it would need to occur after step 1 declare attack and before step 4 apply damage and effects. You can not perform an action during another action, however, as there is no "stack" in Mage Wars. One action has to be fully completed before another can occur.
Thus, the only way you can really utilize this rule is with an Enchantment, since those can be revealed in between the steps of combat or casting. Even they do not really interrupt anything, since you can not reveal them during one of the combat/casting steps, they just have a more precise window in which they may occur by revealing them after one step is completely finished but before the next step begins. You may want to review the rules for enchantments on page 19 of the English Rules v3.
For example: You could reveal the enchantment Divine Intervention to teleport during an attack sequence.
Technically, this rule for cancelling spells/attacks with movement would also come into play if you had something occur during the Avoid Attack Step or the Counter Spell Step that resulted in movement of either the attacker or the target. However, I'm not thinking of anything in the game that currently can do this. I could be forgetting about something though, and of course new things are always on the horizon.
To clarify the rules of the Quick Cast, you will want to review page 8 of the English Rules v3 and also page 11 for more details on actions in general. The most important part though is that you can use your Quick Cast marker before or after a friendly creature's action phase (in addition to the Quick Cast Phases). Thus, you can not use it on your opponent's turn at all, and you also can not use it in the middle of your own creature's action. It has to be used either before or after your creature's action.
Overall though, I think the most important thing for you to remember is that there are no interrupts in Mage Wars. Whenever you do something you must fully resolve it before moving on to something else. You can not perform an action in the middle of another action, and you can not reveal an enchantment in the middle of a step of a phase.