I just don't understand it. How do I approach these kind of problems? Should I create a sheet with methods and note where I should use them?
There are many different ways to approach this question, so I'll try a few of them.
Firstly, there isn't really such a category as "advanced mathematics" as opposed to "not advanced mathematics". Mathematics is like a kind of staircase, with each step building on top of the steps that went before and adding a few new things each step. So if you already at a given step then the next step doesn't seem too hard to climb, but if you are looking several steps up at once it seems like a dizzy height.
Therefore one thing to do when a piece of mathematics seems complicated and confusing is to identify the steps that came before, then go back and review those first to make sure they are clear. To use a kind of analogy, in the US high school system there is "pre-calculus" that is a necessary precursor to "calculus". It is intended that you have all the pre-calculus stuff down pat before you try to learn calculus, otherwise calculus will be too difficult. You can argue the merits of this particular example, but calculus is going to be impossible if you can't do ordinary algebra.
A second response to the question is to observe how mathematics is really all about recognizing patterns, making connections between the patterns, and seeing where the patterns can be applied. This is really hard to do without practice. So doing lots of practice problems on the earlier steps is important to really get them solidified in your mind.
On the specifics of your example problem, one pattern is that "velocity is the first derivative of position wrt time". So if you have an equation that gives the position of a particle as a function of time (you have to pick up that t is time in this example), then differentiating that function will give the velocity as a function of time. A second pattern is that differentiating vector functions wrt time can be done by differentiating each component of the vector individually. A third pattern is that "the component of a vector in the direction of another vector" is a standard pattern in linear algebra. As soon as you see that question you have to say to yourself, "Aha! I know there is a standard way to find that." Then if you can't remember it, go off to your linear algebra textbook and look it up.
It continues with the other parts of the question. Like acceleration being the second derivative of position, or the first derivative of velocity, wrt time. Calling it a "vector acceleration" is kind of redundant; acceleration will be a vector just like the velocity was. "Tangential and normal components" again is a standard pattern. You either know the formula or you don't, but if you don't you can go look it up in the textbook once more.
I hope this helps a bit. Things become less scary as familiarity grows. If you look at something and it seems scary, it tells you that you need more review and more practice. If it seems scary and you have an exam tomorrow you are in trouble. If it seems scary but you are only just beginning the course, then expect enlightenment to come as you work through the problems and practice each concept.
For my part, I couldn't answer that question satisfactorily if I were sitting in an exam today. I don't remember all the necessary formulas. But as I read the question, I feel confident I
could answer it if I went and reviewed the material. That's really the thing; recognizing the difference between not having a screwdriver in your toolbox (but you know where to get one), or not even knowing that you need a screwdriver. Once you know you are going to need a screwdriver the problem is half solved.