How do non-ionic compounds dissolve into water? Into molecules or atoms?
The molecules remain as they are, as discrete particles, they do not break up into single atoms. What can happen to certain compounds when they are dissolved in water is a hydrolysis reaction. But even then, only certain bonds of the molecule are cleaved in a reaction with water, and the molecule does not spontaneously fall apart into the elements it consisted of, because there is no thermodynamic reason to do so.

The same is also true for ionic compounds. Ammonium chloride ($\ce{NH4Cl}$), for example, consists of discrete $\ce{NH4+}$ and $\ce{Cl-}$ ions. $\ce{NH4+}$ is a molecular ion, and it remains as such in aqueous solution and does not break up into single nitrogen and hydrogen atoms/ions. However, the salt can hydrolyze in aqueous solution according to the following equilibrium:

$$\ce{NH4Cl + H2O \rightleftharpoons NH3 + H3O^+ + Cl^-}$$

This is the reason why aqueous solutions of ammonium chloride are weakly acidic.