Start with Inuyasha, Vol. 1 and read forward. Rumiko Takahashi wrote fifty-six volumes as one continuous quest, from the moment Kagome falls down a shrine well to the last fight inside the Shikon jewel, and there is no later jumping-on point. Publication order is the only order.
The decision that actually matters is which English edition you buy. Two numbering schemes are in circulation: VIZ's single volumes, catalogued as Inuyasha, Vol. N and running to fifty-six, and the three-in-one VIZBIG omnibuses, which collect three singles each and renumber accordingly. This path uses the singles throughout. Pick one scheme and stay inside it — mix them and you will either repeat several hundred pages or skip them.
An honest warning about this list before you start
This is one of the more unevenly catalogued manga series we have worked with, and the list below is not the whole run. Fourteen volumes verify to the correct book. Several others return a Japanese-language edition or hand back a different volume entirely, and nothing past volume forty-four could be reached at all. So the volume numbers here jump, and the list ends at forty-four, which is not the ending.
There is a second edition problem inside the fourteen. Seven of the records are catalogued as the Japanese Shogakukan printing or a Spanish one rather than the VIZ English release, so their buy links search by title instead of pointing at a specific edition. Check the language of what you are looking at before you order.
Down the well
Inuyasha, Vol. 1 is the entry point and the only one. A Tokyo schoolgirl is pulled through a well into feudal Japan, wakes a half-demon pinned to a tree, and shatters a jewel that the two of them now have to collect in pieces. Everything the series does for the next fifty volumes is established here.
Volumes two to five are missing from this list, and they are also, conveniently, the weakest stretch of the run — episodic monster-of-the-week material. What you lose is the introduction of Shippo and Miroku, which a synopsis covers in a paragraph. Pick the story up at InuYasha, Vol. 6, where the shard hunt has found its rhythm and Sesshomaru has become a recurring threat, and read straight on to InuYasha, Vol. 7, the first volume where the villain of the week connects back to the central mystery instead of resetting.
Naraku, and the party assembles
InuYasha, Vol. 9 brings Sango and the demon-slayer village — the arc that completes the core group and gives Naraku his first genuinely cruel act on the page. If you read only one volume from this stretch, read that one. InuYasha, Vol. 13 resumes after another gap, with the Tessaiga and Sesshomaru material that turns the brothers' rivalry into a second plotline running alongside the jewel. InuYasha, Vol. 15 moves Kikyo to the foreground and with her the love triangle that readers either stay for or quit over. It is worth knowing which you are before committing to forty more volumes.
The long middle
InuYasha, Vol. 18 turns Naraku's detachments, Kagura and Kanna, into characters rather than obstacles, and gives the series its best subplot: villains who want out. InuYasha, Vol. 20 closes that movement and sets up the Band of Seven, the longest and most popular arc in the series — a defensible stopping point if the length is telling on you. Inuyasha, Vol. 24 has the Band of Seven at full strength, and Inuyasha, Vol. 28 delivers the Mount Hakurei payoff and starts steering toward an endgame.
As far as this list reaches
Inuyasha, Vol. 30 deals with the aftermath of Hakurei and the question of where Naraku keeps his heart, which drives most of what remains. Inuyasha, Vol. 34 covers Moryomaru and the fight over that heart; the plotting is dense here, so read a summary of thirty-one to thirty-three first. InuYasha, Vol. 40 is late Takahashi, with the cast's arcs converging and Sesshomaru finally a participant. Inuyasha, Vol. 44 is where this list ends.
That is twelve volumes short of the finish. To complete the series, look for volumes forty-five to fifty-six as VIZ singles, or switch entirely to the VIZBIG omnibuses and restart the count — but do not mix the two, because the numbers will not line up.
Follow the full ordered path here: Inuyasha Reading Order: Where to Start Rumiko Takahashi's Manga.