I am planning to self-publish my first book under my pen name. Do I need to get copyrights for the book and my pen name before having it published on KDP? Or can I go on without this step?
It may be worth it for you to look into the difference between a copyright and a trademark. While your pen name cannot be copywritten (copyrighted?), it most likely CAN be trademarked.
Your book is under copyright by default.
This may depend a bit based on jurisdiction - the Berne Convention gives the general principles of it all, but different countries (esp. the US) do things a bit differently. In general, though:
Assuming you live in a country that's a signatory to the Berne Convention (you probably do), then copyright exists as soon as you write your story down. You don't need to "get" copyright because you already have it. But in some places (mostly the US, as far as I know) there may be benefits to registering your copyright.
You can't copyright a pen name.
For your book, you can register your copyright under your pen name if you want, but it may have implications for how long the copyright will last. For example, in the US, if you register copyright only under your pseudonym and don't include your real identity, the copyright will last 95 years from publication (or 120 years from creation). If you include your legal name, it will last your life plus 70 years. Different countries have different terms on all this, though.
And, yes, you can publish without registering your copyright. Depending on jurisdiction, registration may not even be available. In the US, registration allows statutory damages, but the cost of the court case that would lead to statutory damages may well be high enough to make that remedy largely theoretical.