Below command can be used for finding both package and current Activity name. Found this very useful command to quickly fetch these two information about an app especially when developing tests with Appium.
adb shell dumpsys window windows | grep -E 'mCurrentFocus'
Response of this command contains both package name and current Activity. For example: in following "com.android.contacts" is the package and "com.android.contacts.activities.TwelveKeyDialer" is current Activity launched on the phone which is connected via adb.
mCurrentFocus=Window{2089af8 u0 com.android.contacts/com.android.contacts.activities.TwelveKeyDialer}
Reference: http://www.automationtestinghub.com/apppackage-and-appactivity-name/