I am wanting to allow people to edit their own task on share point using the edit icon on the left hand side located on the list. What i dont want is for people to edit other peoples information on someone else tasks .
You have 2 options:
A. Use custom permission (using some wf/flow you could trigger it automatically upon creation and grant only the edit rights) but having list with custom permission has its performance drawbacks.
B. You could use the Item Level Security Feature (and I think it will be better option for you) Go to list settings->Advanced settings and there you can set up Read and Edit access only to the author of the list item (in your case Task)
According to my understanding you are looking for Item-Level Permissions.
Below is a good article on how to enable item level permissions in SharePoint. maybe it will help you with your requirements.