Hi @Tom_Comerci, welcome to Asana and the Community Forum!
You are right, you can place tasks you’re working on this week in the Upcoming section and the tasks you’re working on next week or later in the Later section.
Tasks with due dates that have been marked as Upcoming or Later will automatically promote through as they approach their due date. Please note you will need to first move the task to the section, it won’t appear automatically based on their due date. This is a popular request in the forum and we have an existing Product Feedback thread, I recommend you to upvote here: Move task from new to today if due date is set to today 
Please find more information about the auto-promotion feature in this article.
I hope this helps!