While the details vary from camp to camp, generally speaking liberated concentration camps fell under the jurisdiction of the liberating armies, with humanitarian organizations like the Red Cross playing a supporting role. It's important to note that the Nazis often took steps to evacuate camps before the arrival of Allied troops, so liberation did not always take place in a camp.

Typically a quarantine would be setup to control for diseases (as an example, the British VII Corps brokered an agreement with the Wehrmacht that some 19 square miles surrounding Bergen-Belsen would be designated as a neutral zone, with most SS personnel being given permission to leave), and food distribution was managed by military physicians.

The scene from Band of Brothers is fairly accurate, in the sense that prisoners were often kept confined to the camps or were transported to nearby areas (known as Displaced Person, or DP camps). This was for health and organizational purposes. Furthermore, liberation was often meaningless to the prisoners, as thousands more died in the days and weeks following liberation, too malnourished or sick to be saved. Unless carefully monitored, prisoners could easily die from a sudden influx of food. Various methods were used to nurse sick and starving prisoners back to health. Intravenous methods of food delivery were abandoned as the use of injections for torture by the Nazis meant that prisoners often reacted hysterically to this method.

The liberation of Majdanek, Buchenwald, and Bergen-Belsen received the most press coverage, as they were the first camps liberated by the Soviets, Americans, and British, respectively. Soviet press/propaganda in particular attempted to downplay attention to the Holocaust, as it was decided that it detracted from attention payed to Soviet sacrifices. Auschwitz, despite being the largest camp, received very little coverage because of this, as well as the fact that the Yalta Conference took precedence in the headlines.

The typical reaction to discovering the camps was abject horror and disbelief. It was widely known that the Nazis persecuted Jews, and Allied leaders (at a bare minimum) were aware of the camps, and their existence was reported upon in the press. However it was often easily dismissed as exaggerated propaganda. The camps were one of the few instances where propaganda understated the reality of the situation. The scale and depravity of the Nazi crimes were simply far worse than expected. In particular, Eisenhower's reactions are well-documented (and conversely he insisted upon careful documentation of what was found), and serve as a good source of Allied reactions to the camps' discovery.

In terms of further reading there are naturally countless sources both in print and on the internet. I'd personally recommend Hitler's Death Camps: The Sanity of Madness by Konnilyn Feig as a good jumping off point. It's a fantastic overview of the Concentration/Death Camps, with a chapter devoted to every major camp. There's a good summary of what liberation was like, and the chapter concerning Bergen-Belsen is a great case study of the liberation of a single camp (albeit a somewhat atypical one). Let me know if you have any further questions and I'll do my best to answer them.