Is there a way to subscribe to new questions, or new questions with such-and-such a tag?
Yes, you are auto-subscribed to questions you answer or you can click "subscribe-me" button on the right panel.
On the main page you can add or ignore tags in the tag admin block. It has the "Interesting tags" and "Ignored tags" in it. Only works when logged in that's why you probably didn't see it.