I want to notify the users about their assignment list/status of their work items via notification.
Instead of sending email notification, is it possible to show the notifications in Tridion itself? Say for example: having new item like "Notifications" under shortcuts-->mytasks?
Or is there any better way to notify the users apart from email communication, because the users dont want to receive so many mails in their mail box.
Yes. The list of items shown shortcuts is extendable, just like every other list I've ever looked at in the Tridion GUI.
Have a look at this blog post from Jaime to get started. The topic of how to extend is was also covered in this question, but Jaime's tutorial is probably a better starting point.
Once you get some experience with writing this extension, you'll probably run into questions similar to the one Nuno asked here (and that was answered expertly by Jaime and Boris).