This answer is dependent on what kind of music you wish to write. If you want to write solo instrumental music, then you just need that instrument. If you wish to dig deeper and write for multiple instruments or ones you don’t own or play, or you dream of using synthetic sounds, or you just want to record yourself, then you need to be considering using a Digital Audio Workstation, or DAW.
The best approach is to discuss options with your teacher based on your goals, but you can always start very simply and purchase more tools & technology overtime. The teachers will help guide you every step of the way.
The bare minimum to get started would be a notepad and a device that helps you record ideas. (A smartphone or tablet will suffice.)