Are the tasks linked to one another or have you typed in the dates? If you've typed in dates, it won't work. If you have created Finish to Start links between the tasks and updated the Actual Finish of Task 1 - Task 2 should automatically move forward.

If you've typed in the start dates, you've set a start no earlier than constraint -- and you will need to remove the constraints by setting it back to as soon as possible.