道
In Japanese language, this kanji represents the path. As a solo kanji, it is read as みち (michi). As part of compounds, it is read as ドウ (dou).
In English, this is usually translated as the way for various purposes. The essential problem is that this is such a fundamental oriental concept that those unfamiliar with the concept have difficulty grasping the meaning no matter which words are used.
Nonetheless, we can see how the terms are related. The path is the way we travel to get from one place to another. It is the road we walk on. This may be literal, or it may be completely metaphorical.
I simply find the translation of path useful for the kanji on its own, without being associated with anything else, as a good starting point.
茶道
This is chadou, the Way of the Tea (as usually translated).
It is the term used for what we call in English the Japanese tea ceremony.
It is important to note that this is not a Japanese tea ceremony ritual. It is the study and practice of the tea ceremony in its entirety. As such, “the Way of the Tea” seems a better way, to me, of presenting this.
剣道
This is kendou (kendo), as in, the Way of the Sword. This is the martial art involving simulated sword combat using wooden substitutes.
In light of this, even though we are accustomed to saying “the Way,” what is meant is that this is a path that the practitioner follows. The objective is not a particular destination, such as defeat of a particular opponent or victory in warfare; kendo will never provide these things. Kendo is about the experience of walking the path. This is a path to greater self revelation, enlightenment, and character building. It is focused on personal development precisely because learning to slice people in half, for real, is not considered a wholesome family activity.
道教
This is doukyou, or Daoism (also romanized somewhat questionably as Taoism).
Daoism is centered around the idea of the Dao, which in Japanese would be the dou (the “dou” part of “doukyou”). The second part simply means religion. Thus, it is the Religion of the Way.
Having said this, we must still understand this as meaning it’s all about a path. It’s about the way water takes the path of least resistance to the ocean. It is not about ways and means; it is not about methods. It is about the proper path that nature takes, and how we may live in accordance with this path and follow the path of least resistance rather than fighting nature and wasting our efforts.
This is the essence of Daoism.
This concludes the relationship between this kanji and the various ways it is used in Japanese and, through translation, in English.