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For those that do, could you share how you learn(ed) Tosefta? How do you fit it into your learning program of Mishna/Gemara?
Also, what edition of the Tosefta do you use? The ones published at the end of the masechet of gemara (many publications)? If not, what are you using? Which would you recommend?
Any other thoughts in this matter are more than welcome.
As a preface, I have never made a full study of Tosafta on its own. The way I have learned it was to simply look up every single ברייתא that the Gemara quotes. This methodology leads one to have an intimate knowledge of מדרש הלכה and Tosefta. Since every quote has its context, you would then need to learn the surrounding Toseftas and/or מדרשי הלכה. The direct result of this is that you will A) learn Tosefta and B) you will see much more how the Gemara was constructed (sometimes there is an entire amud of Gemara which is simply a block quote of מדרש הלכה, which is the basis for the ensuing discussion).
As for editions: The Vilna edition, which morphed into the Oz Vehadar edition, is fine. But that is all it is. There are better ones than that. One of the classics is the חסדי דוד, Rav David Pardo (he also wrote a commentary on Sifrei). Another classic, written slightly more recently, is חזון יחזקאל, by Rav Yechezkel Abramsky. This commentary is split into the "girsaos" section and the "Tosafos style" commentary. Highly recommended.
However, the absolute best Tosefta BY FAR is the Tosefta KiFshuta, by Saul Lieberman. (I am aware that he was a faculty member at JTS, and many people have remarked that such a Talmid Chacham would work there. There's even a book entitled "Saul Lieberman and the Orthodox" by Marc Shapiro. However. קבל את האמת ממי שאמרו.) There is no better commentary on Tosefta.
This sefer is split in two - The "Tosefta" section, which follows the Vienna Manuscript (with emendations from "כתב יד ערפרט"). It includes brief textual emendations (aka שינוי נוסחות), a brief commentary, and מראה מקומות. Each seder is one volume. The next section is the "תוספתא כפשוטה" part. This is an extremely thorough commentary on every single line of Tosefta, which perforce is a commentary on how both the Bavli and Yerushalmi quote them.
This commentary, as well as the edition of the Tosefta edited by Saul Lieberman, ends at Maseches Bava Basra. The edition of Tosefta, but not the commentary, was completed by Machon Mishor, who printed Tosefta from Maseches Makkos until the end, based on כ׳׳י וינה. (For some reason, the volume on the second half of Seder Nezikin is impossible to find.)
One last edition of Tosefta worth mentioning is the Zukermandl. This was the first "critical edition", based mostly on כ׳׳י ערפרט, with some very brief הגהות. One benefit to this edition is that it is just one volume, and includes all of Tosefta.
EDIT: I have just found out that Tosefta K'Fshuta can be found in digital form on alhatorah.org.
I find learning the Tosefta in conjunction with the Mishnah to be an extremely rewarding experience. The Tosefta often preserves the "raw materials" from which the Mishnah was edited (and these passages can be seen as the earliest recorded version of the oral tradition), and also contains many of the explanations of the terse verbiage of the Mishnah that are incorporated as building blocks into the two Talmudim. Studying the two together heightens both the appreciation of the Mishnah as a literary masterpiece and Rebbi's genius as its compiler, and also allows one to peer beneath the Mishnah's veil, as it were, to uncover many treasures that would have been lost had the Tosefta not been passed down. It is in fact a minor miracle that the Tosefta has survived at all.
I highly recommend Oz Vehadar's Murcheves edition, which contains a beautiful and user-friendly edition of the Tosefta in the back of every volume. It is arranged in Tzuras HaDaf style, with major commentaries surrounding the Tosefta, and additional commentaries (such as Chisdei David) on the margins. It also features Masorat HaTosefta (cross-referencing all of Chazal), Ein Mishpat (cross-referencing the major codes of halacha), and most valuably (for me), variant manuscript readings and the GR"A's emendations. Considering how few people really delve into the Tosefta (sadly), the fact that Oz Vehadar made such a supreme effort to prepare an attractive Tosefta is worth applauding.
In your case, since you already have your own set of Shas, you may want to consider purchasing just the Zeraim and/or Taharos volumes of Murcheves, which contain all of the Mishnayos and Toseftas for those orders (except Berachos and Niddah). What I do is go through the Mishnah at my pace (generally 2 a day) and then once I finish the tractate and it is relatively fresh in my mind, I flip to the back and do the Tosefta for that tractate.
It also is enjoyable to do this in preparation for learning a masechta of Gemara, as you can sometimes predict which direction the discussion will go if you know where the ambiguities in the Mishnah lie, and where the Tosefta clarifies or adds layers to a question arising from the Mishnah.
I do think it's best to learn the Tosefta sequentially, rather than in a subordinate manner (i.e., as though the Tosefta were merely footnotes to the Mishnah), because there is a clear thought process associated with it, and it is very much its "own" work -- Rabbi Chiyya and Rabbi Oshaiah (who are credited with its redaction not only by Chazal but also by the weight of modern scholarship) had their own methodology and flair for juxtaposition, which is easily lost if not read in sequence. And although it can get tedious at times, given its length, there are many gems scattered throughout that will surprise and delight you if you do stick with it.
Rebbi Abbahu descended to Tiberias. The students of Rebbi Joḥanan saw that his face was shiny. They said before Rebbi Joḥanan, Rebbi Abbahu found a treasure. He asked them, why? They told him, his face is shiny. He said to them, maybe he understood a new teaching. He came to visit him. He asked him, what new teaching did you hear? He said, an old Tosephta. He recited about him, a man’s wisdom illuminates his face. J.T. Shab. 8.1.7
Good luck!
Having learned Tosefta independently I can support the other answer which refers to R' Lieberman's tosefta k'fshuta as being the best.
An often overlooked, but very helpful and clear edition is in the back of the old Steinsaltz gemarot.
Lastly, there was (but sadly is no longer) a repository of resources for learning Tosefta online including audio recordings.
Lastly I would caution against using the translation of Professor Jacob Neusner, which like his Yerushalmi translation, did not receive wide scholarly approbation.
