Indian-American. I'm self-taught (though my family has a lot of Sanskrit heritage and my grandfather can understand a lot of it). As I child I learned through the website I linked above, and later on I read Whitney's "Sanskrit Grammar" and MacDonell's "Vedic Grammar", which (especially the first one) provide a concise but thorough understanding of the language.
Reading Sanskrit is good for vocabulary, but IMO writing Sanskrit (e.g. composing your own poems) is much more effective for getting a grasp of the language as a whole.