How do I create a dependency, so that when a user's name is selected to assign them to a task record, other fields in the form automatically populate with the user's email address and phone number?

In QuickBase we call what you refer to as a dependency, a relationship. A relationship always involves two tables.

In the example above one of the tables would be a table of users. The users table would have a field for email address and a field for phone number. The users table, in this relationship, is the master table.

The other table would be the tasks table. The tasks table, in this relationship, is the details table.

To learn more about relationships please read:
https://www.quickbase.com/help/about_relationships.html

Enhance this relationship with "lookup fields" Lookup fields draw additional values from the Master (users) intot eh details table. In this case, you'd just create lookup fields for the email address and phone number fields. (Read how to add lookup fields to a relationship here: https://www.quickbase.com/help/create_lookup_field.html)

Then when you add new task and select a user related master record), their email address and phone number automatically populates.