[Note:] This document is a work in progress, and is expected to change, have references added, links, etc.
This style of martial art was once called Ryukyu Kenpo Toudi JutsuPatM99. This might be roughly translated as Okinawan Kenpo Chinese (Tang) Hand. Karate is the result of Okinawans learning from White Crane Kung Fu (Quanfa) masters, and Southern Shaolin Monks, in Naha parks and while visiting China.
Karate follows templates and techniques, and two person drills, that were then adapted into simplified and new kata. Kata are forms that express what a defender might do in a particular fighting scenario, or with a particular approach or attitude to defence. Kata is the foundation of traditional Okinawan Karatedo.
I started martial arts in the early 1980s with my father showing me Japanese Judo, Nevada. He signed me up with Cottonwood Goju in Utah. Over time, I learned some English boxing, and some Wadoryu from a student of Osaka Toshio. My lineage chart includes Jimmy Bartell's Goju Kyokai, and influences from Taguchi (Shoreikan), the Jundokan International (Chinen), Peter Urban, and Lou Angel of the NCMA, (both students of Yamaguchi of the Japanese Goju-Kai).
Other parts of Karate have different katas, mostly from the lineages of Matsumura, Itosu, and Higaona (Kanryo). Miyagi's Geikesais and Itosu's Pinan (Heian) katas are used for introductory techniques. Sanchin or Naihanchi are used as funadmental katas. The rest go with the system of the teacher. I don't believe in Karate styles unless you understand that every teacher presents a unique style. I believe that Karate katas have original bunkai (practical applications), though the different schools of Karate have different approaches to teaching, and will often give myriad interpretations (oyo), especially with beginners where the more lethal, traditional techniques are reserved.
I like these two videos demonstrating the Geikesai katas:
The primary origin schools of Goju-ryu are Shodokan (Higa), Shoreikan (Taguchi), Meibukan (Yagi), and the Jundokan (Miyatozato). All other existing schools of Goju-ryu are derived from (or influenced by) these. Miyagi did not teach kabudo (Chinese weapons), but the Kodokan (Matayoshi), attached to Higa Seiko's dojo, is closely related. (Shinken Taira is the other if not the primary origin of Okinawan KabudoPM99). The classical katas of Goju-ryu are as followsGH18:
Like most of karate outside Goju-ryu schools, Wado-ryu teaches the five Pinan (or Heian) katas to kyu level ranks. The primary (but not only) katas of Wado-ryu are as follows OS81:
Though it is possible to learn karate from a book, or from internet videos, it is impossible to fully comprehend or master karate fundamentals, or be recognized as a black belt, without a qualified teacher. Karatedo is a life time path and discipline, and a good teacher is necessary to set someone on the same path that they followed. However, for beginners, I recommend the books on Karate fundamentalsST76 from Toguchi (Goju-ryu) and Ohgami (Wado-ryu)OS95.
Find a dojo near you, making it easier to establish a consistent pattern, so training is a life time activity. It's better not to experiment, but find a sensei that works for you, and that you can stay with. Master what they teach. From there the path is perfecting (not perfection of) the art you've been taught, a respect for others and the implications of all martial arts, self improvement, and continuing to glean the original templates and intention of the classical katas. This is the do in Karate.