May 1st: You'll have a school trip to Tatsumi Port Island. This is a scripted event. Enjoy the scenery and dialogue.
May 2nd: Back to school. The 'Strength' Social Link (Daisuke/Kou) and 'Chariot' Social Link (Chie) will become available. Consider starting these. The 'Hierophant' Social Link (Dojima) and 'Justice' Social Link (Nanako) will also become available in the evenings. These are important for story and combat benefits.
May 7th: The 'Sun' Social Link (Ayane/Yumi) becomes available. This is your culture club. The 'Temperance' Social Link (Eri Minami) also becomes available. You can start a part-time job at the hospital to unlock this. The 'Hanged Man' Social Link (Naoki Konishi) will also be available later in the month.
May 12th: Kanji Tatsumi is introduced. He's the next target. The Midnight Channel will show him. This marks the beginning of the next investigation.
May 13th - 17th: Focus on raising your Social Stats (especially Courage for Kanji's dungeon) and starting new Social Links. You'll need to talk to various NPCs to gather information about Kanji before his dungeon appears. Talk to the girl outside the Textile Shop, the boy in front of the bookstore, and the old woman in the shopping district. This will eventually lead you to the 'Steamy Bathhouse'.