Databases are a great resource when conducting research. They offer a variety of media that have been vetted for accuracy and content, so you can feel secure using anything you find in them.
Start with U.S. History in Context and search using the country you are researching with the word immigration (i.e. "Irish Immigration")
Results that come from the U.S. Immigration and Migration Reference Library are what you want to start with for good background information.
This search will give you the best results for this assignment. You can also expand your search into the other databases in this frame.