How to develop abstract thinking and problem solving skills?

Question

I am currently doing post graduation in Industrial Mathematics. During my graduation years, I did not have much trouble in understanding the theorems or I should rather say, the arguments in the proofs (not the actual intuition and insight). Now that I have begun my post graduation, I really want to develop that all essential abstract way of thinking. For instance, while reading Rudin, even though I understand the proofs, I certainly do face difficulties while doing exercise problems. I realise that I have not developed my abstract thinking skills. Apart from the "prove that" type of problems, how important is it to develop problem solving skills for future mathematical endeavours? (like finding the sum of a given series) I have realised that I really like reading and doing math and don't just want to get good grades but actually get a concrete understanding of the concepts, which is, of course, not possible without abstract thinking and "prove that" problems. Any help is highly appreciated. Thanks in advance.