Here is the full list of all 30 Valthirian Arc: Hero School Story achievements.
Finish Chapter 1.
No fallen students for 5 consecutive missions.
Graduate a student.
Enroll 30 new students.
Craft 10 weapons.
Craft 10 accessories.
Graduate 50 students.
Obtain Valthir.
Obtain Crescent Edge.
Obtain Shermanstaff.
Reach Accreditation level A.
All facilities built at Accreditation level A.
Promote a student to Arc Draconus.
Promote a student to Paladin.
Promote a student to Arquebusier.
Promote a student to Harlequin.
Promote a student to Medica.
Promote a student to Scholarsage.
Finish 30 errand missions.
Unite all Five Queendoms.
Ascend Isabel to Mother Seat.
Support one Queendom to rule the Realm.
Rename a student.
Finish Chapter 2.
Finish Chapter 3.
Finish Chapter 4.
Learn an ultimate skill.
Have 1,000,000 G in your Academy's funds.
Buy an item from the wandering merchant.
Unlock all achievements.