I wanted to put in a word that mathematics really is hard and takes time to learn.
In particular, in my experience -- which is, I must say, almost exclusively with pure mathematics, but in many programs in the US the distinction between pure and applied only emerges later on -- relatively few first year math PhD students are reading papers independently. Unless I very much misremember, I did not start reading "serious" math papers until my second year. For what it is worth, I was a student at Harvard, and I entered with a BAMS from the University of Chicago. I was not poorly prepared compared to my American peers. Also for what it is worth, "one-semester graduate real analysis" is what I took as a third year undergraduate. And then I followed it with another semester. And by the way I was a student of number theory. As I recall I spent the first semester of my first year studying for my quals, passed them at the beginning of the second semester, and spent the second semester learning about elliptic curves, local fields and schemes from textbooks of Silverman, Serre and Hartshorne. The idea of plunging into papers without having learned this material: well, it might have added some drama, but almost certainly it would have added to my total time to degree.
I have very mixed feelings when I hear people on this site say things to early career graduate students like: "don't get too bogged down in any one thing"; "you can read textbooks forever; time to start reading papers"; "only spend as much time to learn something as is needed to apply it to your own work"; and so forth. It is not that such sentiments are not applicable in mathematics: I have said all of these lines myself. It is rather that in mathematics this kind of advice gets given out much later in the day: some of it is great advice for mid- and late-career grad students, and some of it sounds more appropriate for postdocs. On the other hand I have seen a lot of students -- including talented ones -- get snagged because they prioritize "their research" over basic learning. I did my PhD thesis on moduli spaces of abelian surfaces with quaternionic multiplication. I didn't know what any of those words meant as a first year PhD student.
Now I write all this knowing that the OP is in applied math, which depending on what that means could either be identical to the pure math experience, wildly different, or anywhere in between. But he is asking about pure math knowledge and seems to have the intuition that it will not come so quickly or easily. I think the most honest, helpful answer is: it does not come so quickly or easily to pure math students at top places. So if you're expecting it to come quickly and easily to you, then you're setting yourself up for disappointment. A certain amount of patient, textbook-driven linear learning will pay immeasurable dividends down the road. How much? Good question: that's what advisors are for.
Well, after all this I may as well take a crack at the precise question asked. Should you solve exercises in textbooks you read in order to gain background on your research? Sometimes. I think that whenever you're reading a math book and get to some exercises you should at least look over them and get a sense of how close you are to being able to solve them. This is an important clue to how much of the material has sunk in. On the other hand, how much time should you spend solving any one "problem set" when you're reading the text in "research mode"? Not very much unless you see how solving that particular problem is relevant to your work (in which case: lots of time, potentially). If you don't know whether the exercises are relevant to what you're doing, you either haven't read closely enough or are reading too linearly: you don't have to read textbooks in order or one at a time. Grab several off the shelf at once. Play them off against each other. Often what you actually need is something that most texts will hint at, drive somewhere near, leave to you as an exercise....but the right textbook will do it wonderfully. Or maybe no one text will say exactly what you want, but together they will. Being able to "triangulate from multiple sources" is, I would say, an intermediate research skill: I know many PhD students who don't seem to have mastered it (e.g. for complete lack of trying!), but it is one well worth developing if you're trying to dive head-first into the literature.