I'm also new to the Fantasy Sports API so this won't necessarily be a complete answer.
You need to be logged in to your actual Yahoo! account (or Authenticated through an OpenID) to see a link to your league page. Your League ID can also be retrieved when you first created the League, or from the !Fantasy Sports site after logging in (for example, my league ID is "97279"):
If you forgot to get it, you can grab the end of the URL when viewing it:
Next you need to plug that information into an authenticated/authorized request. Authentication is done by providing your Yahoo! login or OpenID which has access to your league (i.e. probably only your's for now), while Authorization whether is done via OAuth to specify which aspects of your league can be acessed by who and where that access can happen (i.e. your ASP.net C# app will probably need to use an OAuth library).
In this case though, you're probably already authenticated to your Yahoo! account so the easiest place to work with the data is the console which I see you've already been playing with some public examples for. Then you can start to do more interesting things like drill down to Games/Teams/Players/Stats/odds... although from what I understand, the API is most useful when the league in question has games running, so most requests to your league would be black unless you go back to the previous year's data. The only table I could find for previous years is here:
A request for Baseball stats for an actual MLB player from one of the example leagues from 2010 then, would look as follows:
To see the structure of this URL call, look down below for "The REST Query":
But note to call this from a remote server outside of Yahoo's own fantasy pages or console, you will need to authenticate using OAuth as mentioned above.
That's about all I can add for now, sorry if you know most of this already, but once I've had a chance to develop out a league and test I'll try to add more. I'm sure others who have ran leagues in previous years have had better success too...