The SS (Schutzstaffel) were initially the personal bodyguards to Adolf Hitler. After the Nazi Seizure of power in 1933, the SS was expanded to be the internal security of the regime.
Initially they were a very elite group but eventually expanded due to the on-going war and therefore entry requirements changed as time went by and the war progressed.
To join you would have to meet basic conditions:
The above are what the standard requirements were for an ordinary SS Division. These however changed depending on what part of the SS you are in. For example the Leibstandarte (Adolf Hitler's personal bodyguard) required you to be aged between 17 and 22 and be a minimum height of 5ft 11 inches, later raised to 6ft ½ inches. You also required a traceable Aryan ancestry dating back to 1800 or 1750 for officers.
If you have any other questions on requirements or age limits etc, please do ask.
Your average SS soldier's training regime would not differ much from the training camp that the Heer (German Army) went through. The major difference however would be the sheer volumes of political & spiritual indoctrination that was given to new recruits. A lot of the training was "studying" which in essence was indoctrination. Many children were exposed to this as children and therefore already had this ideology anyway prior to joining the SS, this just enforced it further.
On to your question of how they were morally able to do what they did, as has been mentioned on other threads regarding this subject, the book Ordinary Men: Reserve Battalion 101 and the Final Solution in Poland has some interesting segments. I quote the highest rated comment on a previous topic by a user that has been deleted. This provides a very good answer for your question. I take no credit for it.
As Christopher Browning (author of the popular "Ordinary Men") put it, there were three types of SS men, those who had no issues carrying out the mass killings, those who had issues but used alcohol or other coping methods to carry them out anyways, and finally those who refused outright.
To give you an idea how brutal these killings were, and how much of a psychological toll they took, here is what happened when Heinrich Himmler went to his one and only mass execution:
Himmler became very uncomfortable, very quickly. As the firing squad started, Himmler, was even more nervous. During every volley he looked to the ground. When two women could not die, Himmler yelled to the police sergeant not to torture them
Himmler tried to address this problem by making commanders organize social events and other outreach programs to make the men feel better, yet they were largely ineffective. Here is what a local SS commander, named Eirc Von Dem Bach, said to Himmler about his men's psyche after participating in the mass killings.
Look at the eyes of the men of this Kommando , how deeply shaken they are. These men are finished for the rest of their lives.What kind of followers are we training here? Either neurotics or savages!
The refusal to participate in mass killings was not usually punished. The punishments for refusing to partake in the mass killings would be usually a minor slap on the wrist (a transfer back to Germany,reduction in pay,demotion, small things like that), and that's in the rare cases where was something was actually done. Now soldiers that refused would face criticism from their peers;in the form of people questioning their masculinity and their allegiances. They also wouldn't be looked upon favorably for promotions or any sort of commendation.
Now during the Belzec Trial, there were a few interesting stories that came out, many involving the head of the camp Christian Wirth. Wirth ruled with an iron fist and anyone who dissented was punished. There are multiple stories that came out of Belzec concentration camp. One is that an SS man refused to carry out an execution and Wirth pulled a pistol on him. Or another example, when SS Erich Fuchs was ordered by Wirth to fix shower heads in the newly erected gassing barracks, he questioned Wirth as to why there were showers with no water pipes, Wirth flew into a rage and beat Fuchs; he then ordered two SS guards to take him away and shoot him, they ended up talking Wirth out of it.