UPDATE (May 2021): This is a really outdated answer for old, old Java. For Java 8 and above, see https://stackoverflow.com/a/20906602/314283

Java does appear to be well behind the eight-ball compared to C#. This utility method shows the way to do in Java SE 6 using the Calendar.add method (presumably the only easy way).

public class DateUtil
{
    public static Date addDays(Date date, int days)
    {
        Calendar cal = Calendar.getInstance();
        cal.setTime(date);
        cal.add(Calendar.DATE, days); //minus number would decrement the days
        return cal.getTime();
    }
}

To add one day, per the question asked, call it as follows:

String sourceDate = "2012-02-29";
SimpleDateFormat format = new SimpleDateFormat("yyyy-MM-dd");
Date myDate = format.parse(sourceDate);
myDate = DateUtil.addDays(myDate, 1);

java.time

With the Java 8 Date and Time API you can use the LocalDate class.

LocalDate.now().plusDays(nrOfDays)

See the Oracle Tutorial.

Date today = new Date();
Date tomorrow = new Date(today.getTime() + (1000 * 60 * 60 * 24));

Date has a constructor using the milliseconds since the UNIX-epoch. the getTime()-method gives you that value. So adding the milliseconds for a day, does the trick. If you want to do such manipulations regularly I recommend to define constants for the values.

Important hint: That is not correct in all cases. Read the WARNING comment, below.