Which of Rav Kook's books would be the one to learn if you want to see his learning style and methodology. What I mean is, in which book does he most "show his work" so to speak.
You probably will want to start with Orot. R Bezalel Naor recently came out with a very nice English translation including an extensive historical introduction and notes.
However, be warned that this is a very difficult text to learn alone. The style is very poetic and there are many references to concepts which are not explicit in the text. I learned it with a specialist of R Kook and realized I would not have understood 10% on my own. Audio shiurim going through the text might be a worthwhile alternative.
Another book which helps get into R Kook's works is The Essential Writings of Abraham Isaac Kook by R Ben Zion Bokser which includes selections of various works incl. Orot of course.
I second @mbloch's suggestion for Orot as a book that gives an excellent overview of Rav Kook's hashkafa over a range of topics that were important to him.
However, in terms of learning his learning style and methodology - I would suggest Ein Ayah (עין איה), which shows what went on in his head when he learned gemara. Sadly there's only on Brachot and Shabbat, but both provide a wealth of insight into Rav Kook's thought process, the way he broke down the gemara and analyzed various parts of it.
Note: Though it's a kind of commentary on the gemara - and not just the aggadic parts (it's regarded as a commentary on Ein Yaakov, but there are also non-aggadic sections if I remember correctly) - it's focused on the spiritual and hashkafic aspects and not the more technical halachic aspects.