When you were an undergraduate, studying (I assume) math, the advice you got was good. Read a lot of papers, somewhat superficially, to get an idea of what it is possible to do and how to go about it generally.
But now that you are in a graduate research program you need to do more. The basis you get from textbooks isn't enough to get you to the research frontier in mathematics. It won't get you very close, actually. So, to approach the research boundary of what is known, you need to read (a few) papers fairly deeply. For advanced work, elementary, textbook, techniques may not work. So, you need to delve into the proofs of things to see if there are new and interesting techniques that you need in your own toolkit.
But, as you read a paper there are some things you need to ask yourself, beyond the question whether everything is correct. The big question is whether the paper is complete and is the last word on a topic, or can it, perhaps, be generalized in some interesting direction.
Another big question is whether the techniques used in the paper (not the results themselves) can be used and modified (extended) to solve other similar problems. In this sense the proofs in a math paper may be more important than the theorems. And it is why some new proofs of old theorems are extremely important. The proofs open up new areas of exploration.
Another area of research can be opened up by comparing two papers and examining whether there is a possibility to combine what the two say to come up with something new and interesting. This may or may not require a deep reading of the papers, of course.
To do math research requires insight. Basically, it means being able to find things that might be true but haven't been proven true. Various limitations of logic make this harder than it might seem. So, when reading a paper, another question to ask is "What led the author(s) to think this might be true?" Why did they think it something worth time studying? That can lead to insights.
But, math research, of its nature is narrow and deep. So, you often need to do deep dives into ideas to discover whether there is something hidden but worth exploring. What works for a textbook learner doesn't get the job done for a researcher.
Caveat: This applies to mathematics and some theoretical aspects of other technical fields. I'll make no claim about "technical fields" in general.