Carla:
When using the Timesheet functionality, each resource generates a timesheet
for each period when it's time to report their time. When they generate the
timesheet, the timesheet will automatically pickup their current tasks.
(Note, there are options for generating timesheets with just projects or
with tasks) If you have SP2 installed, the generated timesheets will
automatically display current tasks assigned after the timesheet is
generated, however users have to add these manually if you do not have the
latest updates applied to your system. Finally, yes, tasks will continue to
appear on the My Tasks page. Understand that after each resource enters time
in the timesheet, they must import that data into the My Tasks page and then
submit it to the PM as they have done previously.
There is a free codeplex solution that you can implement to make this a
one-step process as well. Note also that Project Server 2010 offers this
solution baked in, so you need only select one checkbox in the configuration
settings to make this a one-step process in the future.