To enable logging in the driver which you're currently using, select which log types you're interested in and the log level (I'm using FirefoxDriver, enabling all types of logs and collecting all log messages)
LoggingPreferences logs = new LoggingPreferences();
logs.enable(LogType.BROWSER, Level.ALL);
logs.enable(LogType.CLIENT, Level.ALL);
logs.enable(LogType.DRIVER, Level.ALL);
logs.enable(LogType.PERFORMANCE, Level.ALL);
logs.enable(LogType.PROFILER, Level.ALL);
logs.enable(LogType.SERVER, Level.ALL);
DesiredCapabilities desiredCapabilities = DesiredCapabilities.firefox();
desiredCapabilities.setCapability(CapabilityType.LOGGING_PREFS, logs);
WebDriver driver = new FirefoxDriver(desiredCapabilities);
After running the test you can collect the logs, here, I'm solely collecting the DRIVER logs, but you could do this similarly for any type of logs)
Logs logs = driver.manage().logs();
LogEntries logEntries = logs.get(LogType.DRIVER);
for (LogEntry logEntry : logEntries) {
System.out.println(logEntry.getMessage());
}