Are you referring to the out of box approvals workflow? Make sure that
your task  list is set to send the notification  emails. In the Advanced
Settings of the task list: In the "Email Notification" section, make
sure that it's set to "YES".

Yes, I'm referring to the out of box approval workflow. I have
confirmed we are doing that on the task  list.

I can't seem to figure out why the emails  are not being sent to the Assigned to
users. email  notification is set up on the task  list and users are receiving
overdue notices but not the initial email.

Regarding the task list...
I set up the permissions so that the user  can only view and edit  their own task
but I noticed that the task added shows myself in the created  by column . Is
there a way to set up the task list  permissions so the user can only view/edit
tasks that are assigned to him/her?

There is no way to setup the task  list to allow users to only edit  tasks
assigned to them. You need to change the security on each task so that only the
assigned users have security rights to that task. If you are writing the
workflow in Visual Studio you can do this in code, if using SPD you will need to
get a custom action that includes security. You may be able to find a custom
action on CodePlex http://www.codeplex.com .