One bit of advice: There is no shortcut for proper experience. It took me 4 years to come to a point where I can build enterprise level web apps - even though I had the dream of building one immediately, right from the beginning. Start small and build your way up.
Even though I did hate this advice when I was receiving it... Don't try to build the next Facebook platform right now.
Now, to answer your question:
Skills:
You must be absolutely clear about server-client interaction with respect to HTTP. You will never understand AJAX fully without understanding HTTP and behind the scenes of browsers. Note: being clear and knowing everything are two different things. Be clear about HTTP.
Learn about HTML/CSS and JavaScript standards to some extent to know that they bahave differently in different browsers. In the grand scheme of things, they are not that important if you are okay with some framework that handles these for you (I recommend JQuery and JQuery UI).
Learn a little about Linux, Apache, PHP.
How to go about it:
To develop web-apps, you could start with the LAMP stack - Linux + Apache + MySQL + PHP.
First build a small web app that does something trivial - like saving and retrieving user's stuff using AJAX and a nice UI or something. I'd recommend jQuery and jQueryUI for JavaScript and UI frameworks.
Then, build a small web app that just gets data from some Google service, given a user's credentials.. I am not Google expert but I guess Google provides APIs for some services(?).
Then build an app where two people can share their data coming from a Google's service or something to that effect.
Then add your own fancy stuff.
It goes on like that.
If you are a .Net person, you could go with.. Windows + IIS + MS SQL Server + ASP.Net3.5/VB/C#. Guess what? StackOverflow is build on that stack :)
Learning about and using an MVC framework is also a good idea - ASP.Net MVC or something similar for PHP.
Minor clarification - By Google-friendly did you mean SEO-friendly? If so, Google-friendly and web-app don't go well together.
It makes sense to build a Google-friendly website not a web-app.