What is the combat like in 1503? How extensive is it?

Depends on whether you play the single player campaign, scenarios or just custom games. If you are playing a custom game (not sure if that is what it is called) then land based combat takes a back seat. And the only sea combat should be against pirates. The AI players won't bother to attack you until much later, and then only really if you are vulnerable.

The campaign has a lot more of a combat focus, but overall it is still a secondary focus next to the economy, though there are some levels that are purely combat oriented.

I haven't really checked out the scenarios so I can't really say what goes on there.

As for how the combat is? Sea based combat is very simple. You simply equip ships with cannons and tell them attack other ships or targets close to shore. There are trading and military ships of various sizes, with the latter having larger cannon and health capacities at each corresponding size.
Land based combat is more advanced, with various different units types:
http://www.apianto.ch/altes%20Web/Englisch/e_Anno1503/e_Einheiten/e_Milit%C3%A4reinheiten.htm
The land combat itself comes across as pretty clumsy, but it works, kinda. The AI is pretty brain dead and won't do much else but to charge at you. So by using clever unit combinations and effective micro-ing you'll be able to take on a much larger force on land.

Animation and sound wise it as is at best okay and can't hold a candle to a dedicated rts.