Here is the full list of all 15 Summer Pockets achievements.
Participate in Islamon Battles
Complete the prologue
Play table tennis with Tenzen
Complete Shiroha's scenario
Complete Ao's scenario
Complete Kamome's scenario
Complete Tsumugi's scenario
Complete the ALKA scenario
Complete all scenarios
Complete the Islamon Battles
Collect all four Celestial Beasts Islamon
Obtain the title, "Table Tennis God"
Use the Music Gallery
Use the CG Gallery
Use the Character Gallery