If you are dealing with a classification problem like (Yes/No, Fraud/Non-Fraud, Sports/Music/Dance) then use Logistic Regression. If you are dealing with continuous/discrete values, then go for Linear Regression.
The analogous metric of adjusted R² in logistic regression is AIC. AIC is the measure of fit which penalizes model for the number of model coefficients. Therefore, we always prefer the model with minimum AIC value.