Basic smoking money saver calculator
Should I always use objects when coding in java even for a basic program like this one? 


How would you not?  Note that you are using an object in this program now, a Scanner object.  You also use a Calculator class.  Because of course that's how Java works.  There are no methods outside of classes.  

Now, should you make an intermediate class and objects?  Maybe.  Maybe not.  I would like to see more methods here.  For example, consider 

    public static void displayForTimePeriod(String message, double amount, double multiplier) {
        System.out.println(message, amount * multiplier);
    }


That would allow you to simplify most of your code to something like 

        double weekly = cost * packets;
        displayForTimePeriod("In one week you would save: €", weekly, 1.0);
        displayForTimePeriod("In one month you would save: €", weekly, 30/7.0);
        displayForTimePeriod("In three months you would save: €", weekly, 13.0);
        displayForTimePeriod("In six months you would save: €", weekly, 26.0);
        displayForTimePeriod("In one year you would save: €", weekly, 52.0);


You could go further and make a class, but the main thing that it would hold would be the amount.  Note that that stays the same in every call.  

I might add 

        System.out.println("On average, you'll save €" + (weekly/7) + " every day.");


If you wanted, you could make your years calculation more accurate by including leap years.  That would require you to get the current date from the system time.  Remember to handle leap centuries.  2000 was a leap year, but 1900 was not and 2100 won't be.  

I made the monthly number more accurate.  Unless the month is February, there are more than twenty-eight days (four weeks) in a month.  The usual approximation is thirty.  

I fixed the quarterly (three months) number as well.  There are thirteen weeks in most quarters (the exception is the first quarter when February does not have a leap day).  Twelve weeks will be too few for every quarter.