How were they initially fed, clothed and looked after? Whose responsibility was it? Were people taken to hospital or treated else where? Were there any systems put into places to help people find their families or go back home? How did people survive in the long run with no money, belongings or homes to return to?
Everything depended on whether you had a home to go back to or not.
Western European Jewish camp inmates generally returned to their home countries with little problem. However, most of them then faced the enormous task of rebuilding their lives basically from scratch, as they had generally lost many family members and friends, and often their jobs, their homes and their possessions. It often was quite a bureaucratic struggle to regain possession.
Non-Jewish concentration camp inmates generally had an easier time as the situation back home had not changed fundamentally and they could ease back into their former lives. An important exception should be mentioned: Soviet POWs and civilian prisoners were subjected to a severe screening on their return and many were imprisoned, sent to Siberia or even executed on suspicion of "treason" and "collaboration". It is estimated that only one fifth of returning Soviet inmates were allowed to return to their homes immediately after the initial screening. They definitely did not have an easy time of it.
The people you are mainly thinking of with your question are the many Eastern European Jews who were reluctant to return home because of a) antisemitism (especially in post-war Poland there were many killings) and b) there was nothing to go back to as their whole community had been killed. Several hundred thousand Eastern European Jews either stayed on in Germany if they had been liberated from a camp there, or fled to Germany after the war. This might seem odd, but that is where most Displaced Persons Camps were located, which is where those liberated from camps who had nowhere else to go were cared for. The DP camps provided all the basic necessities of life and were run initially by the Allied armed forces, later by the United Nations. These people would spend several years in the DP camps waiting for visas to Western countries, mainly North America, Palestine (later Israel) and Australia. The last DP camp was closed in 1959.
Apart from ex-camp inmates, the DP camps housed a great number of other displaced people as well. Wars tend to have this effect. Most were from Eastern Europe and they inluded both people who had been unwillingly brought to Germany (mainly forced labourers) as those who had in some capacity aided the Germans voluntarily and were scared to return home to (now) communist countries.
The Red Cross played a crucial role (and still does) in collecting information on missing, displaced and deceased persons and in reuniting family members.