How do you develop research skills?
Research is about answering unanswered questions. Primarily those that are important in a field.
So, to get started, ask a lot of questions and keep a notebook. Find out if they are already answered by reading and asking others. Figure out if they are significant in some way or could lead to things that are.
Let the questions lead to hypotheses - what the answers to the questions might be.
In applied math (not my field) the questions might involve things that are particularly difficult in some domain. Ask why? Ask whether they can be simplified. Make a hypothesis about how they might be simplified. Then validate/invalidate the hypothesis. Along the way, other questions will occur to you. Write them down (notebook).
Then...
Ask more questions.