“Goal” is about something important that you hope to achieve in the future and usually something bigger or long-term achievement. It is usually just a noun.
- Set your goals to help you achieve your dream in 2018.设定目标，帮助你在2018年实现梦想。
- Her goal was to get an A grade in her English test.她的目标是在英语考试中获得A级。
- The country can still achieve its goal of reducing poverty by a third.这个国家仍然可以实现减贫三分之一的目标。
“Aim” also means something you hope to achieve but it should be something more short term, also it can be both a Verb or a Noun.
- I am aiming for the champion. (Verb)我旨在拿冠军。（动词）
- what’s your aim? (Noun)你的目标是什么？（名词）
- The main aim of the plan was to provide employment for local people.该计划的主要目的是为当地人民提供就业机会。
“Purpose” is more likely the reason/motive (why) someone are doing something.
- my purpose of this class is to learn about the renaissance era. 我上这门课的目的是了解文艺复兴时期。
- This messages purpose is to teach you the difference.这些信息的目的是教你区别。