R^2 is not a measure of correlation. It is a measure of variability in the dependent variable explained by changes in the independent variables. For example if you have an R^2 of 24% that means your regression model explains 24% of the variability in the dependent variable, which means you are missing 76% of the needed information.
Now you can take the square root of R^2 to get Pearson's r, and that is an estimate of the correlation. But keep in mind that correlation is a very specific type of association, one that is linear in nature. However, you can still have non linear associations, at which point you would either need to add a ploy term to your model, do a transformation or use more advanced options.
A low R^2 means your model is a poor fit, and a low p-value does not mean you have a working model, it could merely mean that what you have is better at fitting the response than nothing at all. You would want to do ANOVA, AIC, BIC, or some other type of model comparison to see if you model works better than a mean only model.
If you want to build a regression model use model selection steps to find a better fitting model, if you merely want to report on the correlation between two variables regardless of whatever else is going on, you can use Person's r as a statistical summary for that measure of correlation. A decent fitting regression model would allow you to explore variable correlation in much more depth, but if your model does not meet the assumptions for a linear model and is a poor fit you will not get reliable information.