DateFormat dffrom = new SimpleDateFormat("M/dd/yyyy");
DateFormat dfto = new SimpleDateFormat("yyyy-MM-dd");  
Date today = dffrom.parse("7/1/2011");
String s = dfto.format(today);

Convert the String to Date first.

Date is a container for the number of milliseconds since the Unix epoch ( 00:00:00 UTC on 1 January 1970).

It has no concept of format.

Java 8+

LocalDateTime ldt = LocalDateTime.now();
System.out.println(DateTimeFormatter.ofPattern("MM-dd-yyyy", Locale.ENGLISH).format(ldt));
System.out.println(DateTimeFormatter.ofPattern("yyyy-MM-dd", Locale.ENGLISH).format(ldt));
System.out.println(ldt);

Outputs...

05-11-2018
2018-05-11
2018-05-11T17:24:42.980

Java 7-

You should be making use of the ThreeTen Backport

Original Answer

For example...

Date myDate = new Date();
System.out.println(myDate);
System.out.println(new SimpleDateFormat("MM-dd-yyyy").format(myDate));
System.out.println(new SimpleDateFormat("yyyy-MM-dd").format(myDate));
System.out.println(myDate);

Outputs...

Wed Aug 28 16:20:39 EST 2013
08-28-2013
2013-08-28
Wed Aug 28 16:20:39 EST 2013

None of the formatting has changed the underlying Date value. This is the purpose of the DateFormatters

Updated with additional example

Just in case the first example didn't make sense...

This example uses two formatters to format the same date. I then use these same formatters to parse the String values back to Dates. The resulting parse does not alter the way Date reports it's value.

Date#toString is just a dump of it's contents. You can't change this, but you can format the Date object any way you like

try {
    Date myDate = new Date();
    System.out.println(myDate);

    SimpleDateFormat mdyFormat = new SimpleDateFormat("MM-dd-yyyy");
    SimpleDateFormat dmyFormat = new SimpleDateFormat("yyyy-MM-dd");

    // Format the date to Strings
    String mdy = mdyFormat.format(myDate);
    String dmy = dmyFormat.format(myDate);

    // Results...
    System.out.println(mdy);
    System.out.println(dmy);
    // Parse the Strings back to dates
    // Note, the formats don't "stick" with the Date value
    System.out.println(mdyFormat.parse(mdy));
    System.out.println(dmyFormat.parse(dmy));
} catch (ParseException exp) {
    exp.printStackTrace();
}

Which outputs...

Wed Aug 28 16:24:54 EST 2013
08-28-2013
2013-08-28
Wed Aug 28 00:00:00 EST 2013
Wed Aug 28 00:00:00 EST 2013

Also, be careful of the format patterns. Take a closer look at SimpleDateFormat to make sure you're not using the wrong patterns ;)

tl;dr

LocalDate
.parse( "2022-05-12" )
.format(
    DateTimeFormatter.ofPattern( "dd-MM-uuuu" )
)

12-05-2022

java.time

Use modern java.time classes. Never use the terrible Date, Calendar, SimpleDateFormat classes.

ISO 8601

Your input conforms to ISO 8601 standard format used by default in the java.time classes for parsing/generating text. So no need to specify a formatting pattern.

LocalDate

Parse your date-only input as a date-only object, a LocalDate.

String input = "2022-05-12" ;
LocalDate ld = LocalDate.parse( input ) ;

To generate text in your desired format, specify a formatting pattern.

DateTimeFormatter f = DateTimeFormatter.ofPattern( "dd-MM-uuuu" ) ;
String output = ld.format( f ) ;

Rather than hardcode a particular pattern, I suggest learning to automatically localize using DateTimeFormatter.ofLocalizedDate.

All this has been covered many many times already on Stack Overflow. Always search thoroughly before posting. Search to learn more.