Hi Karen,
Here is a brief plan for this goal:
1 Create DB collection "Contacts" (type: string, value: string, user(pointer to user))
2 When user adds new contacts - just add new rows(with create REST service) to this collection with fields you need.
When you need get these contacts - just use "list/query" service with where request parameter to filter all contacts with current user.
Also here is a base tutorial that describes how to use create/list/query/delete services: http://devcenter.appery.io/tutorials/...
Regards.