How do I get next month date from today's date and insert it in my database?

You can use PHP's strtotime() function:

// One month from today
$date = date('Y-m-d', strtotime('+1 month'));

// One month from a specific date
$date = date('Y-m-d', strtotime('+1 month', strtotime('2015-01-01')));

Just note that +1 month is not always calculated intuitively. It appears to always add the number of days that exist in the current month.

Current Date  | +1 month
-----------------------------------------------------
2015-01-01    | 2015-02-01   (+31 days)
2015-01-15    | 2015-02-15   (+31 days)
2015-01-30    | 2015-03-02   (+31 days, skips Feb)
2015-01-31    | 2015-03-03   (+31 days, skips Feb)
2015-02-15    | 2015-03-15   (+28 days)
2015-03-31    | 2015-05-01   (+31 days, skips April)
2015-12-31    | 2016-01-31   (+31 days)

Some other date/time intervals that you can use:

$date = date('Y-m-d'); // Initial date string to use in calculation

$date = date('Y-m-d', strtotime('+1 day', strtotime($date)));
$date = date('Y-m-d', strtotime('+1 week', strtotime($date)));
$date = date('Y-m-d', strtotime('+2 week', strtotime($date)));
$date = date('Y-m-d', strtotime('+1 month', strtotime($date)));
$date = date('Y-m-d', strtotime('+30 days', strtotime($date)));

The accepted answer works only if you want exactly 31 days later. That means if you are using the date "2013-05-31" that you expect to not be in June which is not what I wanted.

If you want to have the next month, I suggest you to use the current year and month but keep using the 1st.

$date = date("Y-m-01");
$newdate = strtotime ( '+1 month' , strtotime ( $date ) ) ;

This way, you will be able to get the month and year of the next month without having a month skipped.

You do not actually need PHP functions to achieve this. You can already do simple date manipulations directly in SQL, for example:

$sql1 = "
    UPDATE `users` SET 
    `start_date` = CURDATE(), 
    `end_date` = DATE_ADD(CURDATE(), INTERVAL 1 MONTH)  
    WHERE `users`.`id` = '" . $id . "';
";

Refer to http://dev.mysql.com/doc/refman/5.1/en/date-and-time-functions.html#function_addtime