Hi @Julie_Ghostlaw welcome to the community.
I will assume you are using the Free Version of Asana. There are a couple of features in Premium that would assist you… Start Dates, Custom Fields & Advanced Search.
The first thing I think you are not making the most of the My Tasks View, this has the “New Tasks”, “Today”, “Upcoming” & “Later” sections. The reason I say that is that you comment of “Sub-tasks are hidden (and when I assign it to myself, it create a separate tasks under a new section, which is inconvenient”. When you assign a Task to yourself it immediately appears in the “New Tasks” section, it is not a seperate tasks it is the subtask that you assigned to yourself.
Suggest as step one have a look at this Guide article on My Tasks view;
What I would look at doing is to start using one or more projects to help categorise the work that you are doing to make the following up easier. Some options would be An Email Project, Phone Calls, Your Superior, Waiting On… That way you can have the Specific Task assigned to a relevant category/project. You can then use Sections in those projects rather than Custom Fields to provide a process status. You can also use Multi Homing of Tasks if a task is relevant to more than one category/project. Have a look at this article for more detail on projects and multi homing.
Now for the Follow ups, I would use the Due Date as being the date that you want to follow the piece of work up. That way it will appear in your “Today” section in My Tasks when it is due today, you can then follow the person up and then reset the Due Date to the time you are going to follow them up. Just make sure that you then move that Tasks to Upcoming or Later so that the Auto Promote function works.
By using Projects when you complete a Task it won’t be visible in the project any more but by changing the Filter to be just “Completed Tasks” you will be able to see all of the Tasks that you completed for that project.
As a final thing I would suggest you have a look at this Community post and the example of implementing “Getting Things Done” in Asana as I have feeling this maybe a good way for you to progress and move your work through.
Hope this is of help and definitely ask questions of the Community as there are lots of people willing to help.
Regards
Jason.