What are some Samurai cultural pursuits and a primary source (photograph, painting, document or extract from document) that can explain their pursuits?
As currebell pointed out, the Samurai class were experts on the finer details of beauty. Tea ceremonies, calligraphy, elaborate armour and sword detail and poetry were all important to the Samurai.
This is an excerpt from the website for the Asian Art Museum's Lords of the Samurai exhibition:
The samurai culture and code of conduct, bushido, have long captivated the imaginations and aspirations of young and old in the Western world. More than just professional warriors, Japanese samurai of the highest rank were also visionaries who strove to master artistic, cultural, and spiritual pursuits.
Lords of the Samurai takes an intimate look at the daimyo, or provincial lords of the warrior class in feudal Japan. The Hosokawa clan, powerful military nobles with a 600-year-old lineage, embodied this duality of fierce warrior and refined gentleman.
The exhibition features more than 160 works from the Hosokawa family collection housed in the Eisei-Bunko Museum in Tokyo, and from Kumamoto Castle and the Kumamoto Municipal Museum in Kyushu. Objects on view include suits of armor, armaments (including swords and guns), formal attire, calligraphy, paintings, tea wares, lacquerware, masks, and musical instruments.
Should you need primary source evidence of the Samurai class' hobbies and past-times, this website could be a good place to start.
Yabusame was a sport practised by Samurai which required precision of archery technique to hit stationary and moving targets while riding a horse. Inuoumono was a form of traditional Yabusame which used dogs as moving targets. There are some great books and websites about this subject, too, many of which contain primary source evidence.
It may seem odd, given the warrior-like nature of the Samurai, but the practice known as "cha no yu," or tea ceremony, played a significant role in the culture.
Tea ceremonies, rituals of the elite, were both simple and elaborate. The tea room was entered into only by a very small door through which on had to crawl, symbolizing humility. Swords were left outside. Once inside the hut, a scroll with a simple message, chosen by the tea room's owner, was visible. Guests read the greeting or proverb and sat around a hearth. A light meal that was both visually pleasing and tasteful was served, followed by Saki and usually fruit.
The tea itself is the last thing to be served during the ceremony. The guests exit the tea room. The host replaces scroll is replaced by a vase with a single flower and prepares the tea. Guests return and are served. The host and guest both strive to create an atmosphere of peace and tranquility.
The Frazier Museum had an exhibt of Samurai culture including artifacts from tea ceremonies. You can read more at the link below.