(and a list is exactly what I've been wanting...)
First Heinlein I read was "Puppet Masters", which got me hooked.
I recommend three Heinlein books to start: Starship Troopers, The Past Through Tomorrow: Future History, and The Moon Is A Harsh Mistress.
Starship Troopers isn't very long, but I think it's the clearest and most straightforward introduction to Heinlein's philosophy and writing style; it's a clean, undecorated blade, bright and shiny and durable, and, oh, yeah, very sharp. This book is on the USMC's reading list because of the questions it makes you ask.
The Future History is a collection of shorter fiction (some short stories and a longish novella), perfect for the beach. It's an expansion of the Heinlein you meet in Starship Troopers, spread from his very first published science fiction short story, "Life-Line," to "Requiem," to my personal favorite, "If This Goes On ..."
The Moon Is A Harsh Mistress demonstates how to manage a modern revolution, and is a precursor in style and substance to much of Heinlein's later works --- eyes-wide-open realistic cynicism about the ways of the world, and competent, idiosyncratic, grown-up people hell-bent on changing said world into something better.
Rah, rah, RAH!