All of the above days are pretty heavy on cutscenes, centering around Medjed and your next Palace. When you gain control at night, in your room, you can't do anything but go to bed. On the 22nd, Ryuji actually has a good idea and says to ask around the neighborhood. Just walk around and speak to whoever has a speech bubble above their heads. Make sure you check the alleys and one clue you get is that Sojiro buys a lot of food. Luckily for you, there happens to be a supermarket in the area, so check there. There will be more scenes that round out the last two days, with a lot of background as to what's going on. Note that both the Justice and Judgement Confidants will advance a rank.
It looks like your plans have been set in motion, although one piece in getting into the next Palace is missing. Before you set off for Sojiro's house, you will have the Magician Confidant rank up to six, making it so you can craft all of the infiltration tools now. When you are asking questions, pick the bottom three options to find out the last keyword for Futaba's Palace. Once you finally regain control, head inside of the pyramid and despite the need for exploration, you really can't go anywhere. Some more scenes will happen and you'll end up leaving the pyramid for now. Back in your room, feed your plant some Mega Fertilizer (three points in Kindness), then head to bed.
Your mission for Futaba's Palace has begun today, with a deadline of August 20th for the entire operation. Also, if you look at your calendar, you will notice that you are now on summer vacation for the remainder of July and all of August, so there will be no more school. Rather than dive straight into a new Palace, it's time to go back to working on the usual, but don't worry, you will be going into the Palace in a few days. For starters, answer Makoto's text and hang out with her.
Confidant: Answer "He sounds suspicious" for the first option, then "I got this" for the second one.
When you get home, look at your IMs and answer Chihaya's to go and hang out with her.
Confidant: Pick "You're such a hard worker" for the first answer, then whatever you want for the second and third ones.
Note: If you used up any Lockpicks in Mementos earlier in the month, you can call up Kawakami and have her make some for you (it'll cost 5,000 yen and she can only make one each time). If you wish to get all of the treasures in the next Palace, you're going to need three of them.
There's no reason to hang out with Yusuke today, despite him visiting Leblanc, so just ignore him and head for the Underground Mall. Find Ann and depending on your relationship with her, either invite her to your room (dating) or go to Shibuya (friends). At night, go to the Diner and order a Hot Coffee to get three points towards Knowledge and one point for Guts.
Note: At this point in the game, the second-hand shop near Leblanc, where you bought the TV, seems to be selling a Retro Game Set. Purchasing this will allow you to play games in your room.
Ryuji will be hanging out at Leblanc today, but just like with Yusuke, ignore him and go on with your day as normal. Go to the Underground Walkway and hang out with Yusuke here.
Confidant: Pick "Do you want me to strip?" as your first response, then "I'm sure you will" for the second one.
It's time to hang out yet again with Chihaya tonight, who will text you, so respond to the IM to save a little bit of money. Note that Mishima will text you late at night, but you won't need to accept his offer.
Confidant: Choose whatever you want for the first three responses, then "You're just Chihaya to me" for the final one.
Today, you're going to be maxing out another Confidant, this time it will be Ann's. Respond to her text to go and meet up with her, unlocking another hangout spot on the map, Seaside Park. The responses don't matter here, as you will rank up with any choices. Not only are you able to fuse Ishtar, the most powerful Persona of the Lovers Arcana, but Ann's Persona will also change to Hecate. It will learn Evade Ice and a couple of resistances will change (it will null fire now and become strong against electricity).
At night, you will have to IMs from Ohya and Kawakami, but since it's raining outside, go ahead and study at the Diner in Shibuya once more (Hot Coffee) to gain three points in Knowledge and one in Guts (your Knowledge should level up to rank 4 at this point). Note that you will receive an invitation from Makoto in bed, to see a movie. Unlike the past few invitations, take Makoto's offer.
You will see the movie with Makoto today, earning you three points in Guts and ranking it up to level 4. When she asks you a question, answer it with "Not at all" to gain some bonus points towards her Confidant. At night, answer Chihaya's text to hang out with her, although you won't rank up.