I am developing a web/mobile application.
It is a list and task manager/personal organizer.
I would like to keep it completely free for users.
Moreover, I don't want to show any advertisement on the public area of the website (I don't want to push people off before they sign up).
And it has became my job as I recently resigned from my fulltime job in a company.
What I am doing at the moment is displaying some Google Adsense in the account of the user (after they log in).
I am still waiting for the first payment.
I have the feeling that I could do better. I mean, joining better advertisement networks.
I suggest you try to do the exact opposite of what you mentioned in your question "I would like to keep it completely free for users". Charge money for it.
If the web app gives something of value to your users then just charge for it. You can always offer a "light" version of it for free and with ads.
Charging money to users of your web application is "normal" and you should feel comfortable with it. Yes many people might argue that competitors offering their apps for free will take your users, but then again, how many times do free products really give you what you want/need?
I think advertising will be a challenging route to create a fulltime revenue from a web app. I would be surprised if even the likes of Evernote or Spotify actually make much of a profit from their advertising.
You may want to consider either a freemium route or selling mobile apps to compliment your main free service.
Not directly related to your question but, you also need to consider what features will attract users to your service over competitors like RTM.
Are their any niches you could dominate where you could be the best? i.e. the best task organiser for [insert hobby or profession here].