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" then is a good place to start. However, it is still very large, and organized in a way that is ellusive to the uninitiated. So, the following selections — in the order given — are suggested to establish a foothold in the Buddhist scriptures as the basis for further exploration (through following the footnotes provided on DhammaTalks.org). 

Toehold

I've selected this smaller set of suttas as most essential for an introduction to Buddhism. 

"Setting the Wheel of Dhamma in Motion" (SN 56:11)

"The Noble Path" (AN 4:237)

"The Five (Brethren)" (SN 22:59)

"Chewed Up" (SN 22:79)

"The (Fourfold) Round" (SN 22:56)

"Desire" (SN 1:69)

"An Analysis of Dependent Co-arising" (SN 12:2)

"With Kāmabhū  (On the Cessation of Perception & Feeling)" (SN 41:6)

"The Noble Search" (MN 26)

"An Analysis of the Path" (SN 45:8)

"Unbinding (Nibbāna)" (Ud 8:1—The Nature of Unbinding) 

"Sāriputta" (Ud 4:7)

"Unbinding (Nibbāna)" (Ud 8:2—On Seeing Unbinding)

"Perceptions" (AN 7:46)

"Unbinding (Nibbāna)" (Ud 8:3—The Existence of an Unfabricated Dimension Allows for the Escape from Fabrication)

"A Being" (SN 23:2)

"Unbinding (Nibbāna)" (Ud 8:4—Unbinding as Independence)

"Alone" (SN 36:11)

"Aflame" (SN 35:28)

Toehold for Philosophers

The suttas of this small collection highlight apsects of the dhamma that philosophers—especially those of a Socratic bent—need to realize. 

"Love" (AN 4:200)

"Attached" (SN 22:53)

"The All" (SN 35:23)

"For Abandoning" (SN 35:24)

"Siṁsapā Leaves" (SN 56:31)

Expanded Foothold 

This is an expansion of the initial toehold above. These reinforce the general aspects of the dhamma, and address misconceptions that to me seem particularly relevant for Anglo-Americans in the twenty-first century. 

"The Shorter Analysis of Action" (MN 135)

"Penetrative" (AN 6:63)

"To Yamaka" (SN 22:85)

"To Gandhabhaka" (SN 42:11)

"To Mahāli" (SN 22:60)

"Clinging" (SN 22:121)

"Sister Cālā" (SN 5:6)

"The Longer Discourse to Saccaka" (MN 36)

"The Great Causes Discourse" (DN 15)

"Aggregates" (SN 22:48)

"Clinging" (SN 12:52)

"To Koṭṭhita" (SN 35:191)

"Short Set of Questions and Answers" (MN 44)

Large Foothold in Compassion

Though a small set of suttas, this selection greatly expands on the above—providing a very large foothold indeed. The theme here is compassion—and "keeping it real." The dhamma is not about a perfected and "blessed" lifestyle. The Buddha suffered—even after gaining knowledge of his release—even on the threshold of his total unbinding, the entrance into permanent nibbāna. And all those who follow the noble path suffer. They struggle with comprehending the dhamma. And they especially struggle with putting the dhamma into practice. 

"Vakkali" (SN 22.87)

"Total Unbinding" (SN 6:15)

"Nodding" (AN 7:58)

"Paṭācārā" (Thig 5:10)

Repetitive Forms

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")