Interestingly enough, it was very similar to that of the allies. There is only so much deviation of running, shooting, obstacle-courses etc that you can do. The variable to note here however is the fact that United States wasn't at as much pressure to push out troops of basic training as Germany was especially during the later parts of the war. As an example, within the Kriegsmarine, u-boat training deteriorated closer to the end of the war, and that combined with better radar, made u-boats rather ineffective. Mostly this was due to time concerns, as many men on uboats died.

a pretty good link in regards to Waffen SS training. http://www.globalsecurity.org/military/world/europe/de-waffen-ss-heer.htm

I can't really talk too much about SS training comparably, but there was from what I understand a lot more indoctrination. I would understand that the physical requirements were also a bit tougher, at least for the officers.