How to most effectively get good at a new topic? I have recently had a change of interest and would like to focus on a different area within my field. That being said, I want to stress that the area happens to differ quite a bit from my current area. I am not too far in my studies, so it's not like I am changing my PhD thesis midway or anything like that.

However, I would like to know how I can most effectively get to grips with my new topic. I am obviously not completely new to it, but I have so much more knowledge in my old topic and am now worried that it will be a disadvantage, because I might be competing with people for PhD spots that have had their focus for many years and thus much better writing samples etc.

Any advice on how to become good relatively fast?
The simple answer is: Review Lectures, then read the referenced books / papers (in this order).

There are a variety of good lectures on free course websites out there. (For example on youtube or coursera.) It'll give you a good introduction and - more importantly - will tell you what is relevant to know about in this field.

If you can't find a lecture or at least slides, try to find a few (recent) PhD dissertations in the field and read their introductions / related work / background chapters. These are usually available for free and they will give you a good overview of what is considered relevant.

You could then move on to study recent research papers, but - depending on your prior knowledge - you may not know half the methods that have been used here.

In any case, find out which are the most relevant conferences in this field and see which Professors publish there a regularly. (The Prof. is usually last on the author list). From there you'll hopefully find the most key people in the field and check out their websites and latest publications to see what they are working on.

I'll have to stretch that the right approach will largely depend on how well you are familiar with the basics of the field already. You may have to study a few books or some additional mathematics course work relevant to this new field you are interested in.

Since you appear to be going to university - maybe it's also a good idea to just ask the Prof. most relevant to this field on campus for references and guidance. They will most likely know who's book and publications to read. Maybe this should actually be the first thing you want to do ...