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