The Buddhist textual canon is vast, having been composed over several centuries by various sects. It can be difficult to know where to begin the study of this literature. One approach is to begin at the historical beginning. The texts of early Buddhism are best preserved in the Three Baskets of the Theravada tradition. The first division is the "Discipline Basket," which concerns Theravada rules of monastic life. The second division is the "Scripture Basket," containing discourses of the "Awakened" — that is, the "Buddha" — Siddhartha Gautama, and discourses of his close disciples. The third division is the "Higher Teachings Basket," which delves deeply into the analysis of sentient existence, and minutest details of the liberation from suffering. The "Higher Teacings" assume a background in the other two divisions, and "Discipline" is generally not immediately relevant for non-monks. So, these do not make the best introductions. The "Scripture Basket," however, makes for an excellent introduction. 

The following provide a good foothold in the "Scripture Basket." 

"Setting the Wheel of Dhamma in Motion"

"The Five (Brethren)"

"Aflame"

"Sāriputta"

"The Noble Search" 

"Short Set of Questions and Answers"

"With Kāmabhū  (On the Cessation of Perception and Feeling)"

"The All"

"For Abandoning"

"To Gandhabhaka"

"Aggregates"

"Sister Vajirā"

"The Water-Snake Simile"

"To Kaccāna Gotta"

"To Ānanda"

"To Anurādha"

"The Great Full-Moon Night Discourse"

"Clinging"

"The Great Total Unbinding Discourse"  

"Siṁsapā Leaves"

You will note that the suttas are very repetitive in form and content. One reason for this is that the texts were transmitted orally during the early period of the emergence of Buddhism. In some cases, they were chanted by monks in a rhythmic and melodic fashion. The following is an example of this tradition being practiced today in the original Pāli language. 

Audio Recording: Dhamma-cakkappavattana Sutta (Setting the Wheel of Dhamma in Motion)