I am currently studying in a graduate finance program and have an undergrad in accounting. I am interested in learning programming as I think that in finance, accounting, and Economics, it will be a major skill to have. I was told to follow the data science track, and some say Machine learning. Can anyone tell me which track is best for quantitative finance or financial economics? What are the differences between Machine learning and Data science, and can any other courses help me?
What is your end goal? Do you want to build algorithmic models or do you just want to learn Python and SQL for data retrieval, cleaning, and analysis? I suggest taking a look at the syllabuses (syllabi?) for both and determine how they’d fit with your goals.
DS is more general and has an overview of what DS is, the processes involved of finding, obtaining, cleaning, analyzing data (hypothesis testing, statistics), and eventually building models, this includes learning Python and all its libraries and learning SQL. It also includes a small slice of NLP and ML (supervised and unsupervised model building).
The ML path is more focused on learning Python and the math involved w/ML. There are overviews of learning different levels of ML models (in 3 parts).
But, before diving into that I’d recommend learning Python. I mean, you’ll learn it in both paths, but it’s more focused in a Python 3 course. I feel like ML is very popular right now but so many people jump into it and get overwhelmed b/c they don’t have any experience in programming. I’m not saying that one has to be a Python expert to understand the concepts presented in the DS & ML courses, but one should have a basic understanding of it or have a programming mindset. The Python 2 course is free, or there are several other places to learn it for free (YT) just to get an intro to it.
Everyone learns differently, so what works for some might not work for others. My answer is subjective b/c I learned data analysis, then focused on Python and DS (I don’t regret it either). Any reply here will be just that: subjective. 
I’ll add that much of the coursework is shared in these paths and once it’s completed in one area, it’s marked as complete in other areas.