How do you determine the products from the reactants?

In chemistry, a lot of it is memorization and experience.

For the first one, NaOH is formed. The key thing to note is that it's formed in water. NaOH is a strong base so it completely dissociates in water, leaving you with Na+ and OH- ions. This reaction is a basic acid + base reaction (double displacement). What ends up happening is that you have to figure out whether your products are soluble or not (by memorizing your solubility rules).

The second one is again a knowledge question. Disodium Trioxide is what's called an anhydrous acid. What that means is that when you mix it with water, you get one of your common acids (nitric acid in this case). This is also acid/base chemistry and is usually expected for you to remember because it's what acid rain is.