The American Dream is commonly associated with certain ideals, such as hard work, independence, faith, freedom, self-government, prosperity, and family. You could research primary documents (such as diaries, journals, and editorial articles) from colonial days through the present day that articulate the American Dream, written by people from all walks of life. Often, these pieces will reflect many of the above mentioned ideals. Many colonists came to America from various countries around the world (primarily in Europe, initially) for the ideals of freedom (of speech, press, education, worship, assembly, etc.) and the opportunity to succeed. America is often called the Land of Opportunity.
The American Revolution centered around the idea that the colonists did not want to be under the strict control of a king far away. The colonists rebelled against Britain because they didn’t want to work for an unjust government which did not allow them to prosper from their own work. A pervasive philosophy among most Americans is that if one puts their mind to a task, works hard, and has freedom to succeed, then usually they will do well. People from all social levels and backgrounds can become educated in America and rise to the very prestigious positions in America, despite many hardships along the way. There are countless personal stories you could research and use to illustrate people who exemplify the American Dream (Frederick Douglass, Helen Keller, Abraham Lincoln, Ralph Lauren, etc. )
You can explore another philosophy that is commonly associated with America and that is one of working for wealth. Many Americans and people around the world associate America with unending opportunities to succeed and amass money, as well as material goods. In this way, an ideal of hard work can relate to a materialistic philosophy. A person can value hard work for the work and satisfaction alone (idealistic), while another may value work only for the wealth or goods one can gain from the effort (materialistic). Either way, since America is so vast with so many resources, usually people can find work.
You could analyze another example of the American Dream: owning land. In many countries around the world now (and in the past), only the powerful or wealthy can own land. Colonists came to America for the opportunity to own land, as land was more expensive and scarce in Europe. In America, people of varying social classes can purchase land, whether many acres or part of an acre. Some Americans just want to own land to own a piece of the earth (idealistic). Some Americans want to own land to work the land and make a living on it for their families. Other Americans want to own a great amount of land to sell in real estate transactions or work for tremendous profit, such as oil, hunting or fishing rights, etc. (materialistic).