On the Case Dashboard, you will see the option to "set status" next to the case title at the top of the screen. You and anyone with whom you are sharing this case controls the status, so simply select the appropriate stage and you will see the case timeline change accordingly.
You can sort "My Cases" on your main Dashboard by status, and you can filter "My Cases" to see only, for example, your "Open" cases.
Note that you can not reopen a case that has been archived, so if you think that there is a possibility that you will come back to a case, you may want to keep it as an active (i.e. non-archived) case with a status of "Open," "Resolved - Partially," or "Closed - No Resolution."