What are the basic rules used to determine the solubility of a solid mixed in water?
The first step would be to determine if the solid is covalent, metallic, or ionic. If it is metallic, it will not dissolve in water.
If it is covalent (two or more nonmetals bonded together) you must then determine if it is a polar molecule by creating a Lewis Structure and looking for asymmetrical, unbound electrons on central atoms. If the molecule is non-polar it will not dissolve in water. If it is polar, it will dissolve in water because of water's polar nature following the general rule: "like dissolves like".
If the solid is ionic you must then apply the rules for ionic solubility to see if the substance will dissolve in water. There are many tables available to determine solubility. See reference below.