Is there any way we can look at the chemical formula of various compounds, do some calculations or something and figure out which compounds will have higher solubility as compared to the others.
It depends on the solution and the solute. Usually, the rule is "like" dissolves "like". A polar solution will dissolve polar compounds and a non-polar solution will dissolve non-polar compounds.
As far as I know, to find the solubility you have to determine it experimentally. This can be done by "feel" first. If you have a compound that doesn't have many polar groups on it, chances are it can be dissolved by a non-polar solution. If you know what the structure looks like try to match it with a solvent that looks similar to it.
As a partial answer to this broad question, one may provide a prediction about the partition of an organic compound between water and octanol, $\log p$. This often serves as one of the preliminary checks (prior to chemical synthesis) of compounds foreseen for a pharmaceutical application among the Lipinski rules. An example like this or this may provide a first experience; some molecular editors offer such an estimate, too.
As usual, as such a prediction is based on a model, there are shortcomings; like i) how to treat naturally ionic compounds, or ii) how to account for amino acids, for example. While it may guide between more or less likely water/octanol soluble compounds, an experimentally determined value is preferable.