It is interesting to note that, in general, the early movement of plants and animals was not equal in both directions. Though many Old World plants would eventually come to the New World, it does not seem that as many came over early on. In addition, they generally were brought by the settlers for their own use and did not spread as much to native cultures.
For example, we can see that many New World plants made it to the Old World and became important parts of the diets of people in those areas. Perhaps the two most famous of these are the tomato in the Mediterranean and the potato in Northern Europe. It is hard to imagine the diets of those areas without these imports from the New World. There is not really a similar record of Old World plants becoming part of indigenous diets in the New World early on. This may be in part because few native civilizations really survived contact with the Europeans. Their civilizations were, instead, either destroyed or assimilated into European civilization, as with the natives of what became New Spain.
With regard to animals, the Americas had relatively little to offer. There were many European animals that came to be important in the Americas very quickly. The most obvious example of this is the horse, which completely transformed the culture of Native Americans in the Great Plains. Cattle, sheep, and pigs also changed New World societies and diets to a great degree.