How does solubility vary with different solvents and different temperatures?
The simple answer is
In general (but not always) an increase in temperature should result in an increase in solubility WITH RESPECT TO THE SOLID SOLUTE. That is, if you pump more heat into the the system, the solvent should have more energy to disrupt solute-solute interactions and dissolve the solute. This is practised in the technique of recrystallization. The bulk material goes up into solution at higher temperatures, and it crystallizes at lower temperatures.
And as to solvents, well you know the general rule that like dissolves like. Polar solvents generally dissolve and polar and ionic materials.
The capacity of a solvent to hydrogen-bond, makes the solvent a good candidate to dissolve and solvate ionic species. This is why water is such an excellent solvent. On the other hand there are a host of ionic solids that are almost insoluble in water, because the ion-ion interaction is too strong to allow ion separation.
If there is a more specific issue, raise it, and I will try to discuss it. This is an experimental issue, and what works works.