There are two distinct accounts of Creation in Genesis: the famous six-days account in Gen 1 (culminating in the creation of the Sabbath in Gen 2:1-3), and then a retelling which focuses on the Creation and Fall of Man, in the rest of chapter 2 and chapter 3. Sort of, Gen 1 shows events from a panoramic view, then Gen 2-3 focus in on particular events.
Now, scholars have identified four strands of narrative in the Torah/Pentateuch. What precisely these represent is hotly disputed; that they exist, is not. (They may be four completely independent stories of Hebrew origins, shuffled together; they may be what was preserved from Moses's teaching by four separate groups, later merged back together, etc.) Gen 1-2:3 is from what is termed the Priestly ("P") tradition, one fascinated by genealogy and ritual and with a focus on God's transcendence, and with a tendency to use "YHWH Elohim" (rendered "the Lord God") to identify Him. Much of the "frame story" uniting the various individual stories in Genesis is P's work. On the other hand, the Adam-and-Eve story, like the majority of the other compelling stories in the Torah, is the work of the Jahwistic ("J") writer, whose characteristics include: 1. He's a simply fantastic storyteller. 2. He has a strong focus on the patriarch Judah and events in the South of the Holy Land, the area that was assigned to the tribe of Judah. Most scholars believe that the J narrative was preserved there. 3. His normal usage for identifying the deity is YHWH, God's unspeakable name, which most translations render as "the LORD" (with 'LORD' in caps.). The other traditions use either YHWH God or avoid mentioning YHWH as a name until He reveals it to Moses. (The other two main traditions are E, using Elohim for God and focused on Joseph and the Ephraim area in the North, and a tendency to show God acting through nature; and D, responsible for most of Deuteronomy, and with a strong prophetic, ethical focus.)
Out of respect for the sources, whoever united or reunited the four strands into the Torah as we have it tended to leave in quite a few contradictions of detail and "doublet" preservations of about the same story. To give an example of how this works, take the Parting of the Red Sea. The bit where Moses holds up his arms with outstretched staff is vintage P; J tells of God parting the waters and the familar story of the Israelites crossing dryshod and then the Egyptians becoming mired and engulfed in the returning waters; in E's retelling, he makes sure we know that it was God sending a strong wind which pushed back the waters.
Anyway, details about whether plants or people came first (or even whether it was one "day" (yom, which might mean either '24-hour day' or 'notable period of time', as in usages like "in King Arthur's day") or six, were left un-"corrected" out of respect for the traditional sources. If one reported it one way and the other the other, this was not at issue -- the important thing was not whether God did it on Wednesday the Fourth Day of Creation, but that He's the one that did it. Coming at the Creation stories from a modern Judeo-Christian vs. secular scientific stance, we miss the crucial point the writers were at pains to make -- All of Creation was created by God Himself, and He called it good. The old "spirit=good/matter=bad" dichotomy, and the idea that God Most High musta worked through assistants, angels and such, dates back at least to the days that Scripture was being assembled into the Tanakh, and the writers are at pains to make clear that it was God Himself who did it and who called it good. This, not the details of what he happened to do when, seems to have been a primary emphasis.