Also, applied mathematics programs teach proofs because they want to show that their applied methods are guaranteed to work, as opposed to being fragile, ad hoc guesses.
* Machine learning / deep learning in particular
* Modeling financial data
* Game development, creating a physics engine
I've done all three in a humble amount.
So definitely calculus, probability/stats and linear algebra. Maybe I'm missing something. I've been able to get by without calculus and linear algebra somehow.
My recommendation would to pick the one you're most excited about and focus on it. You mention in another comment that you prefer to work with other people and not do books/MOOCs alone. In this case I suggest that you find a graduate student in the relevant area and ask him/her to tutor you (i.e. give you an independent course in the topic). This might cost $50 to $100 USD per hour if you shop around (perhaps more). The goal would be to learn enough of the basics that you can pick up the more advanced things yourself.
Good luck.
I would just take these 3 courses then, starting with linear algebra.
See how much less humble you can proceed after that.