I use Wunderlist to manage tasks with my assistant. We both LOVE it. We have a shared task list that I can put items in along with notes, attachments, etc. We can leave comments back and forth if needed and I am alerted as to when its done.
As for email, I do not give her access to my email but instead either (1) forward emails with instructions if necessary or (2) respond with her added to the 'cc' so she can take it from there. Personally, I prefer to see all mail that comes in and it takes just a few seconds to forward and pass it along. I should clarify that she responds on my behalf but doesn't pretend to be me and responds from her own email.
The other thing she does is manage my calendar through google calendar. We established conventions for adding new meetings so that I'd always know who I'm meeting with, background info, etc. I gave her my preferences for best times for meetings, locations for meetings, including travel time, etc.
Lastly, we use Dashlane to manage passwords and credit cards. I create the 'entity' in Dashlane and then share it with her through their system so she can log in or pay on my behalf but doesnt know what my password is.