You wrote:
I want to say the following sentence with fewer words:
- I am learning in order to keep earning and to improve earnings.
[a] version I have in mind:
- Learning, to keep and improve earnings.
My first thought is: No, don't do it. If you are trying to shorten a sentence, don't do so by removing the sentence's subject and predicate. Leave the I am in the sentence.
My second thought is: It would help to know what you are trying to say. Your original sentence is quite ambiguous. For example:
What are you learning about? Are learning more about personal finances? Or do you mean that you are obtaining an education, which will increase your earning potential?
What do you mean by "improve earnings?" Are you in business, and you want to learn how to make more money each year? Or do you work as a taxi driver, and you are trying to learn how to get bigger tips by socializing better?
What do you mean by "keep earning?" Does that mean you are trying keep your job? Or that you are trying to keep making the same amount of money in your job?
These may seem like petty questions, but it's nearly impossible to give hints about a revision if we don't understand the original.
By the way, your original sentence employs parallelism, which is an effective device when talking about two related concepts in a single sentence. From what I've seen so far, your efforts to shorten your sentence are only ruining something that was a decent sentence to begin with.