They all end up with the word 愿, meaning merely "wish", so we can try to focus on the first character.
志 is short for 志向, which is related to "job" and "direction", namely, what kind of JOB you would wish to accomplish, in addition to the implication that you are HEADING TO or will HEAD TO achieve it. Resolve is required if you use this.
Ex: 我的志愿是做医生。(Doctor is a JOB)
意 means "mind", namely, whether you are WILLING TO do A or do B,C,D,etc. You are confronted with finite choices if you are using 意愿. You would also use it when you want to say "I wish to do A INSTEAD OF B". Again, resolve is required if you use this.
Ex: 我的意愿是做医生。(Instead of a lawyer, etc.)
心 means "heart", and is simply talking about what you wish to happen. This one focuses on your FEELING or INCLINATION, so is usually translated "desire".
Ex: 我的心愿是做医生。-> I desire to be a doctor. (because sth touched my heart)