The area map has several stages which must be cleared in order. Parties and equipment can't be changed on the area map. You will not be able to return to the home screen until you complete all stages or give up. Losing on a battle will also return you to the home screen, losing all progress and energy used unless lapis is used to revive.
There are two types of stages, one being battle stages, in which wave battles occurs, and the other being scenario stages, in which story scenes play out.