Here are three ways to set a default assignee to a newly created task.
1) Via a task form
Task forms are customized templates that can be used to create tasks that have specific attributes. From Product settings → Task forms, you can create a form, such as "Bug Report," that will set the default attributes for any task created via that form. In addition to specifying the default assignee, forms also allow you to set the default List, Priority, Status, and even the description!
Once saved, you can use that form to create new tasks from the the new task modal. As shown below, the newly created task will have the same attributes that were selected in the form -- such as a default assignee.
2) Via "Auto-fill" in the new task modal
The "Auto-fill" option in the new task modal allows you to create new tasks with pre-defined attributes. When auto-fill is turned on, once you assign a user, any new task created from the modal will set that user as the assignee by default.
*NOTE: the attributes populated when "auto-fill" is turned on are based on the attributes of the most recently created task.
3) Via your "Assigned to me" list
If you want to set yourself as the default assignee for newly created tasks, simply create the tasks in your "Assigned to me" list.
Similarly, if you want to set your teammate as an assignee for a newly created task, simply create the task in their "Assigned to me" list. To do this, search their name, click on their user list, and create a new task.