I'm interested in hearing how people have used Google's tools, in particular Tasks and Calendar, in the context of Getting Things Done or a similar system.
I actually use Gmail/GTasks for my own customized GTD methodology, and it really helps me get things done. Here's some tips:
tl;dr: I basically did this but in Gmail.
The new gmail design, which makes Tasks much more accessible, makes a GTD/Gmail combination extremely easy to set up. Hope this helps!
For GTD, have you looked at ActiveInbox yet? I used to use it when it was called GTDInbox or some such. It is not bad, though if you have large volume of emails, your mind just keep getting pulled aside by that magic Inbox (6) sign.
Gmail tasks I do not think are particularly useful for GTD, though you could always have them as Next Actions. I use Calendar as Tickler and trying to use it as hard landscape.
The biggest problem is Project Support. You would think that Gmail would make perfect Project Support since the email is right there, but actually you need to be able to edit the email and/or extract actions out of it. I tried replying to the emails to myself with actions, but that was not comfortable either.
So, in summary, if you live in GMail, it is a good place to start organizing, but - I feel - to get higher levels of mastery you will separate your email flow and your Project/Action spaces.
This is all obviously GTD stuff. I don't know how to do other systems. I do know that RememberTheMilk integrates with Gmail as a FireFox plugin sidebar, but when I went to try it, it was breaking with each new Gmail update. Actually, GTDInbox was having similar problems.
What I use are a set of rules for managing my email and getting things done.
I use Rules to better email from SSW which provides a robust set of rules that work very well in Gmail. Instead of "delete" I use Archive.