I can see where you would be unsure about where to start with this. It is a big question.
I suppose that the biggest success for the US was keeping communism from spreading to any areas of the globe that were really vital to US interests. Communism did not reach Western Europe, for example, or the Middle East or Japan. It did not have any success in Mexico. These are the places that were most important to the US. Since this was one of the major US goals in the Cold War, you would have to call that a success.
There are a few things that you could call failures. You could say that the "loss" of China was a failure. So were the Vietnam War and the attempt to get Cuba back from Castro. Another major failure, you can argue, was the US's inability to convince various "Third World" countries to ally themselves with the US. This was particularly true of the Arab nations. This failure to create friends for the US was also a major failure and one that has outlived the Cold War.
Seeing that this is for an assessment, I think that you have to turn to what was explored in your class and by your instructor and course readings. Honestly, this is the first place to look. Whatever opinions you will get from here should supplement these. At the same time, I think that the question itself will open many doors of perception. Be ready to have to sift through varying opinions. I think that one definite success of the Cold War was that the liberal democracies of "the West" were validated to a great extent. Western style representative democracy and free market practices became emulated by nations controlled by the former Soviet Union and became exported all over the world. The reality of globalization only happens because of the success of the United States in the Cold War. I would say that this would constitute one of the successes in the Cold War. Even if it was not an intended or directed success, the fact that the United States did not end up losing the Cold War is one of the reasons why this ends up becoming one of the successful elements of the conflict. I think that one of the failures that emerged out of the Cold War was the inability to think critically about global threats to individual freedom. Being mired in the paradigm of Cold War mentalities made it difficult to envision a post- Cold War world. For example, the arming and support of Bin Laden and Saddam Hussein in order to advance Cold War aims precluded the understanding that these individuals could turn out to be global threats later on down the proverbial road. Embroiled in the mentality of the Cold War, there was a full embrace of how the global threats to freedom and democracy would look when a Soviet presence was absent. This myopia became one of the elements that helped to form an amorphous foreign policy in the 1990s and caused a level of surprise in the attacks of September 11, 2001.