It depends how complex the game you want to make is. You can easily learn some basic PHP and javascript inside of a month (like pretty much any other language), but it'll take a lifetime to master (like any other language).
I would recommend you start out small - plan to implement just a subset of your features, and take them on one at a time.
I would plan to use libraries / frameworks. For Javascript I would currently recommend JQuery because I find it easy to use, it has a large community and it's well documented. Using a library like JQuery allows you to easily ignore a lot of the browser specific details, since they deal with all that nonsense for you.
For PHP I personally use Zend Framework - this is a massive beast that isn't always the best documented, but if you start with the "Quick Start" tutorial and then only look at features as you need them, you shouldn't get overwhelmed too easily. One of the great things about Zend is that you can pick and choose what features you want to use without hassle.
With regards to servers, yes you can use a web host and bypass setting up a full server yourself very easily and cheaply. You can find local installations (eg. XAMPP) that will allow you to quickly set up a local install of Apache, PHP and MySQL to get started with.