Subtasks can be added to your cards via the checklist function.
When opening a task, you can add your checklist items at the top, using the + checklist icon.
Each of the items can be assigned to someone, and the assignee will receive email notifications on that assignment, just as they do for a task card assignment.
To assign a checklist item to a team member, to edit, delete or move it up or down the list, hover your mouse pointer over the item, and the respective options will show up.
You may also be interested in trying Task Dependencies - a clever way to connect tasks in a hierarchical manner.