
Your main objective when exploring dungeons is to steal the treasure. The treasure is guarded by the palace “boss” so you have to make sure that your party is adequately leveled and equipped before facing it. While it’s not possible to clear an entire castle within one visit, you need to make sure to reach the farthest safe room you can to get the most out of your infiltration. Pushing forth without adequate supplies is just suicide.
Take note that after stealing the treasure will make the palace collapse and will prevent you from entering it again. That said, make sure to loot all chests, including the locked ones before defeating the palace boss.
Here's the usual run
1st run: Up to the mini-boss area
2nd run: Up to the treasure location (securing route)
3rd run: Final run to the boss (after sending the calling card)
When you exit a palace after an infiltration excursion, you’ll get an infiltration log. This will feature the milestones you’ve accomplished for this particular infiltration attempt. And since you can fast-travel to any safe rooms, you can immediately start close to where you left off. Due to the limited time of the game, it’s imperative to get as much stuff done as possible during your visits, preferably the second run to secure the route, then finally leave a calling card (necessary for the treasure or the desire to materialize) and face the boss.

Since combat is the only way to increase your character levels, it is strongly recommended to clear palaces thoroughly - like picking hidden items with your third eye and defeating all enemies, as long as you can manage your resources. Enemies respawn whenever you change floors or areas that require loading. That said, do not overstretch your team too far that your escape will be cut off since you’re too weakened or out of supply to return to a safe room.
Since palace visits can be limited without sacrificing your other After School time slots, you need to manage your team’s HP and SP stock wisely. You can’t just go around using special attacks even on weak enemies. Exploit their weaknesses and use all-out attacks to take out larger groups of enemies. Consider talking to shadows as well in an attempt to get them to join you, give you money, or random items. This is a great way of earning additional resources especially when dealing with Persona that you already have.

Furthermore, SP is very valuable here when exploring palaces. While there are powerful physical attacks that consume HP instead of SP, SP is needed for using healing, curing, and revival skills. There’s also a wide variety of elemental attacks that use SP primarily which is crucial for exploiting shadow weaknesses. Once your party’s SP is depleted and you don’t have any way to replenish them, clearing a palace will become increasingly hard.

Always buy the necessities, make the necessary tools, and buy equipment upgrades before entering a palace. There are no shops or resupply points so careful preparation is important to prevent wasting precious time and infiltration attempts. Make sure that you also prepare a day before your planned infiltration since you can only make valuable tools during a free evening. Also, don’t forget to sell the treasures you have stolen from the castle to earn more funds.
Also, consider fusing or executing personas before starting the infiltration so that the bonuses from any rank-ups or new confidants that you unlock during your daily life activities will be applied to the latest fusions you’ll perform.
Whenever you infiltrate, always have at least several lockpicks. The first dungeon alone has four locked chests. Locked chests usually contain really valuable items compared to what you can normally find in ordinary, unlocked chests. Having them in your inventory ensures that you get the valuable loot and use it as early as possible.

Another thing worth taking is a Goho-M. This invaluable item allows you to return to the castle entrance no matter where you are in the palace - perfect for scouting runs where your main objective is to push your party to the limit to get as deep in the palace as you possibly can.