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There are a growing number of contemporary translations of the Buddha's discourses, or suttas, as well as many commentaries. In reading a sutta, please take your time. Allow yourself time to reflect on what is being said and notice the questions that come up in your mind. They can help you look more deeply into the meaning of the sutta. It is often helpful to read more than one translation of a sutta to get a clearer sense of its meaning. Here are some resources to help you in your exploration of the suttas.
Before beginning, you may want to read John Bullitt's excellent recommendations for reading the suttas: "Befriending the Suttas."
Bhikkhu Bodhi's In the Buddha's Words: An Anthology of Discourses from the Pali Canon is an excellent place to start in studying the suttas. It has selections from many important suttas along with Bhikkhu Bodhi's commentary on the Buddha's life and teachings.
Bhikkhu Bodhi's translations of the suttas are available in print:
The Middle Length Discourses
The Connected Discourses
Numerical Discourses of the Buddha (both as an anthology and as the complete set)
Bhikkhu Bodhi also has audio commentaries on The Middle Length Discourses.
Maurice Walshe has a translation of The Long Discourses of the Buddha in print.
John Ireland has translated The Udana and the Itivuttaka.
The Sutta-Nipata, the oldest collection of discourses in the Pali Canon, is available in a translation by Ven. H. Saddhatissa.
The Dhammapada is available in a number of translations. Gil Fronsdal has translated it. Access to Insight has translations by Acharya Buddharakkhita and Thanissaro Bhikkhu. And Ajahn Munindo has produced a "Dhammapada for Contemplation".
Thanissaro Bhikkhu has translated many suttas from all of the collections mentioned above. He has made them available for free through the web site Access to Insight.
If reading a collection of the discourses, it is generally recommended that one begin with the Middle Length Discourses and then read the Long Discourses.
Mindfulness: The Direct Path to Realization by Analayo Bhikkhu is a highly recommended contemporary translation of the Satipatthana Sutta (The Four Foundations of Mindfulness) and commentary on the sutta that combines scholarly comparison with practice concerns.
The Sutta Readings web site offers "a library of free audio recordings of English translations of Pali suttas" so that you can study and enjoy them in their original oral, and aural, form.
Access to Insight is highly recommended as a resource for many articles and books on the Buddha's teachings.