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 not relevant for non-monks—at least not immediately. 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 the following of 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 Kāmabhu  (On the Cessation of Perception & Feeling)" (SN 41.6) 

"The Noble Search" (MN 26) 

"An Analysis of the Path" (SN 45.8) 

"Unbinding (Nibbana)" (Ud 8.1—The Nature of Unbinding) 

"Sariputta" (Ud 4.7) 

"Unbinding (Nibbana)" (Ud 8.2—On Seeing Unbinding) 

"Perceptions" (AN 7.46) 

"Unbinding (Nibbana)" (Ud 8.3—The Existence of an Unfabricated Dimension Allows for the Escape from Fabrication) 

"A Being" (SN 23.2) 

"Unbinding (Nibbana)" (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) 

"Hindrances" (AN 9.64)

"Food (for the Factors for Awakening)" (SN 46.51) 

"Attached" (SN 22.53) 

"The All" (SN 35.23) 

"For Abandoning" (SN 35.24) 

"To Unnabha the Brahman" (Abandoning Desire by Means of Desire, SN 51.15)

"Simsapa Leaves" (SN 56.31) 

"Eight Individuals (1st Version)" (AN 8.59) 

"Deathless" (The Four Establishings of Mindfulness, SN 47.41)

"The Establishing of Mindfulness Discourse" (MN 10) 

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 Mahali" (SN 22.60) 

"Clinging" (SN 22.121) 

"Sister Cala" (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 Kotthita" (SN 35.191) 

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

Large Foothold in Compassion

Though not great in number, this subselection of suttas massively expands upon the above—providing a very large foothold indeed. The theme here is compassion—or, in twenty-first century American lingo, the theme is about "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. They especially struggle with putting the dhamma into practice. For the liberation from all struggle and suffering, a convolution of devotion to the dhamma with devotion to the Buddha is often essential succor. And there is always comfort in the Sangha—that vast throng conveying the dhamma throughout the world, simply by letting their lives, in so many peculiar ways, be shaped by those exquisite noble truths that were in some way first discovered and taught by the Tathagata. 

"Vakkali" (SN 22.87) 

"Total Unbinding" (SN 6.15) 

"Eight Individuals (2nd Version)" (AN 8.60)

"About Cunda" (Ven. Sāriputta’s Passing Away, SN 47.13) 

"Gems" (SN 2.1) 

"The Great Total Unbinding Discourse" (DN 16) 

"Treasures" (Khp 6) 

"To Vacchagotta on Fire" (MN 72) 

"Going for Refuge" (Khp 1) 

"About Golissani" (MN 69) 

"Nodding" (AN 7.58) 

"Patacara" (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")