Is there a way I can set up the Stack Exchange app (for Android or iOS) so that, when there is activity, I receive a notification?
Now I only receive one if someone answers or comments on my question or answer, or they "at" me. I want more.
You could get an RSS notification app, such as this one, and then point it at Mi Yodeya RSS feeds you're interested in. You can find links to them at the bottom-right of most pages.
https://judaism.stackexchange.com/feeds - New questions
https://judaism.stackexchange.com/feeds/tag/shabbat - Recently active shabbat questions
https://judaism.stackexchange.com/feeds/question/12760 - New activity on our big Samoan question
The Android and (I understand) iOS apps notify you for inbox events -- comment and chat pings, and new answers to your questions. I'm not aware of a way to get it to produce other notifications, but you might want to ask on Meta.SE.
I use a combination of the app and the mobile web site on my phone; I find the app very hard to navigate, and the mobile web site lets me see the list of active questions easily. The mobile web site also shows me my inbox and rep changes, so in the end I use the app to get my phone to buzz at me when Stack Exchange wants to tell me something. I'm presumably missing some important capabilities there. :-)