Hello Damian, I would suggest "The Blind Watchmaker" first. It's content is not as dense. It is a better introduction. However, "Climbing Mount Improbable" would probably be the best for the beginning student of his books.
I believe that beginning with the selfish gene is the best way to start off with Dawkins. His Best book so far according to Dawkins himself is the "Extended Phenotype", and to appreciate this work, according to me, needs a conceptual foundations from the selfish gene. Moreover, "The Selfish Gene" itself is a book full of ideas and commonsense. It gets you to see the evolution at the level of genes (if you had not before), which I guess is the central dogma for most of his works. Well, most of his works are brilliant and they make a good reading and can leave you pondering over so many things, its a pleasure. So you can read any book of his I guess, but if I were you, I would start off with the selfish gene.
Furthermore, The Selfish Gene is a classic. The other mentioned books (Blind Watchmaker and Extended Phenotype) are also excellent books but were written to popularize and to extend The Selfish Gene. A newer edition of TSG received the addition of the chapter Nice Guys Finish First, in my opinion the best introduction to game theory and by itself worth the book.