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:
Your inbox is, well, your inbox. Keep one email address (or forward your multiple email addresses to it) and have it contain all schedule confirmations, new project ideas, etc. Process this to zero once per day every day. Checking your email too much (or spending too much time on stackexchange ;)) is a guaranteed way you won't get anything done.
If you come up with good ideas, email them to yourself. For example, I am currently working on my personal website -- if I come across any pages that I would like to reference in the future, or come up with any ideas, I email them to myself. Because I have good labels, I can refer back to those ideas in the future when I am actually coding/designing/whatever, and the Gmail space quota is large enough where it really doesn't matter if you have a few extra emails lying around.
Labels are your friend. Filters are your best friend. If you can take all that information you are receiving and have the computer automatically organize it for you, that's less work you have to do. Create labels that have meaning and then create filters that will detect and organize labels with that kind of information.
Google tasks allows you to have multiple lists. You can make lists of all actionable items, and create Tasks from your emails. When I begin a project, I'll generally make a list of steps I have to take, and throw them into a list. I can then make my next actions a variety of tasks from the different lists I have to do, and I can set milestones by the percentage of tasks I have done.
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.
http://www.sergeykin.com/2015/07/en.html - here is my implementation of GTD on the basis of Google Calendar