How can I predict what dissolves what?
As you note, there are some general rules about solubility. For example, acetone is a highly polar organic molecule, so it's not surprising it has high solubility/mixibility in water. The "like dissolves like" rule is generally based on polarity and size/shape considerations, and it's a fairly good rule.

But solubility is complex because you have two big sources of driving force:


enthalpy: The molecular and intermolecular interactions between the solvent and solute are favorable. Consider methanol dissolving in water: it forms hydrogen bonds with the water, so the enthalpy is favorable.
entropy: This is often the larger driving force. You create a far more disordered result with a mixture than with two pure components.


While most chemists develop reasonably accurate intuitions of enthalpy, the entropic effects are harder to predict.

Many efforts have been made to predict solubility, both qualitatively and quantitatively. The comments above give some good links. Consider that this problem is important to chemical synthesis broadly, but also to drug delivery (i.e., can a drug be given orally and dissolve in the stomach, can a drug go through the blood-brain barrier?)

Perhaps the most successful is the Abraham general solubility model which attempts to predict solubility of organic compounds in 84 different solvents. Here’s the article.

Basically, it breaks down solubility into multiple components (grabbed from Andrew Lang and Jean-Claude Bradley):


E is the solute excess molar refractivity in units of (cubic cm per mol)/10. It represents the solute's polarizability and gives a measure of the ability of a solute to interact with a solvent through n- and $\pi$- electron pairs.
S is the solute dipolarity/polarizability. It gives a measure of the solute's ability to stabilize a charge or dipole
A is the overall (summation) hydrogen bond acidity. The hydrogen bond acidity descriptor measures the extent of hydrogen bonding by the solute in a basic solvent. 
B is the overall (summation) hydrogen bond basicity. The hydrogen bond basicity descriptor measures of the extent of hydrogen bonding by the solute in an acidic solvent.
V is the McGowan characteristic volume in units of (cubic cm per mol)/100.


The nice thing about this model is that it makes sense -- you want a solute that matches lone pairs, hydrogen bond donating or accepting ability, polarizability, and volume for the solvent.

The catch is that surprisingly little data is available. The late Jean-Claude Bradley encouraged his students to make accurate solubility measurements, but also to publish them on the web. In this way, they gathered a large spreadsheet of "compound X dissolves Y amount in Z solvent."