增进
增进 is used to describe an improvement in friendship(友谊)，feelings(感情), understanding(了解)
我经常和同事一起吃饭，增进感情。
I often eat meals with my coworkers in order to improve feelings.
It directly translates to improve feelings but it means improve and build coworker relations.
我去汉英学校，所以我能增进对中国文化了解。
I go to Sinology Institute so I can improve my understanding of Chinese culture.
提高
You use 提高 to describe levels(水平), grade(成绩), awareness(知识), price(价格), and quantity(质量).
我需要学习中文。我想提高我的汉语水平。
I need to study Chinese. I want to improve my Chinese level.
通过“冰桶挑战”活动，人们提高了对ALS的认识。
Through the ice bucket challenge, people have raised awareness for ALS.
改善
改善 is used to describe improving the quality of something. This word is more formal. 改善 Relations(关系)，Air quality(空气质量)，Health(健康)
北京市政府一直努力试图改善空气质量。
The Beijing government is consistently and seriously trying to improve the air quality.
进步
进步-This word is intransitive. You can’t say you 进步 something. It is used for studies, work, and ability.
三个月内，我的中文进步了很多。
In three months, my Chinese improved a lot.
他学习汉语已经三年了，可是从来没有这么大的进步。
He’s already studied Chinese for three years, but he hasn't made this much progress
上个月，他的工作有很多进步。
Her work has improved in the last month.
