What ways can I purchase a newspaper subscription for viewing on an iPad? I understand there's a Newsstand app. I am wondering, however, if I were to purchase a newspaper subscription through the Nook or Kindle systems, will I be able to read them in the standard iPad (iPad 3, iOS 6) apps for each publisher?

One reason for concern is that years ago, I found that some things I'd purchased on the Nook -- public domain books at least -- we're invisible to the Nook Windows app. ( Newspaper edition purchases may too have been unreachable).
For the Kindle system, if one looks carefully under the product description such as you would find under this list, http://www.amazon.com/Kindle-Newspapers/b?ie=UTF8&amp;node=165389011,  one can see a link for "Available on these devices". This link will tell you if an Amazon subscription for that newspaper can be read on "Free Kindle Reading Apps",  or exclusively on "Kindle e-readers".  

In order to have the freedom to read on multiple devices (which seems to be the intent of your question, but forgive if I assume too much), you'd want to choose a newspaper available on "Free Kindle Reading Apps".