[SalesForce] Help in converting string to datetime in correct format

Please use parse() method instead of valueOf() method.

Datetime dt = Datetime.parse(myDate);

Please refer for more details on DateTime Methods: https://www.salesforce.com/us/developer/docs/apexcode/Content/apex_methods_system_datetime.htm#apex_System_Datetime_parse

UPDATE:

I've provided the code for you which I tested.

DateTime startDT2 = DateTime.now();
system.debug('select dt2>>>'+ startDT2);
String myDate = startDT2.format('M/d/yyyy h:mm a'); 
system.debug('select mydate>>>'+ myDate);
Datetime dt = Datetime.parse(myDate);
system.debug('date_string >>>'+ dt);

The result is 16:18:50:041 USER_DEBUG [7]|DEBUG|date_string >>>2014-12-09 10:48:00

You must use an ISO 8601 date, specifically something like:

2016-10-30T15:27:02.000Z

Which is October 30th, 2016, at 3:27 pm (+2 seconds) in Greenwich Mean Time.

To parse dates in the user's locale, use the appropriate form for the user. For example, in the United States, we would write:

10/30/2016 3:27 pm

Carefully note the order of month, day, and year, and pay attention to the separator (/). You need to input the date exactly as salesforce shows dates to you, because it is locale dependent.