It is claimed that the order of events in Genesis 1 and 2 are different. Here is day-by-day the creation order in Genesis 1:
Genesis 2, however, might seem to place humans before plants or before animals. Here's Genesis 2:5-7:
This seems to say that man was created before plants, but strictly speaking it only says that plants were not yet "in the land" or had not "sprung up". An anonymous reader of this page suggests that this could mean they existed but in some sort of dormant state, having not yet been "planted"; though this is hardly the most natural reading.
Another suggestion is that plants "of the field" refers specifically to cultivated plants. This argument has some weight, since the passage notes that there was nobody "to work the ground" - i.e. there were no farmers yet.
Another problem is Genesis 2:18-19:
This appears to place the creation of man before animals, yet "the LORD God formed every animal" could simply mean "God created new specimens of creatures already in existence", though this is not the natural reading. Alternatively, the ESV and some other Bibles solve the problem by describing this creation in the past tense:
This way, God had already created these creatures. I'm not an expert in ancient Hebrew and can't say whether this translation is legitimate or not.
As you probably know if you're reading this, I generally argue against infallibilism. In this case I strongly suspect the two Genesis accounts had different authors with different understandings. However, I feel that this view is not really proven by a nuanced reading of the text.