> That being said, should I conquer the two provinces to the north or fortify my one measly little province?

In a word, yes. (that is to say, try to do both)

If you leave Denmark unguarded that seems to raise a flag for computer opponents to launch a surprise attack, so keeping that southern frontier guarded will help you to stay at peace with the Holy Roman Emperor.

At the same time, you'll never make enough money from a single province to support you for long, so you really have no other choice but to expand, and Norway and Sweden are actually great provinces for an ocean-trader power like the Danes to conquer.

> Add on top of that the fact that I'm not entirely clear on how to establish trade routes with other nations and it's pretty easy to see that no matter which path I take (conquest of the two northern territories or no) I quickly go into debt.

Over time you'll begin to develope a better sense of what structures and units are cost-effective in your early game strategy. You definitely have to spend wisely because there isn't a lot of room for error.

With the Danes my view is that you need to start out building a land army at least up through the point that you control Sweden and Norway, but after that you should start thinking about ships (especially Longships which are less expensive for you than other types).

Get your ships strung out into trade routes leading into the North Sea and beyond, and get trading posts and ports set up in Scandanavia, and before you know it you'll be rolling in dough. My present Danish campaign with 8 provinces is cranking out a profit of around 10,000 florins a year - a lot more than I can spend.

> How are you supposed to keep the Danes afloat?

Trade. Get your trade structures built, and establish a contigous path of ships to friendly/neutral foreign ports.