Learn German: Ich öffne das Büchlein, um ein Blümchen zu finden. - I open the little book in order to find a little flower.

Whenever you add the diminutive suffixes -chen or -lein to a noun, the new word is always neuter, regardless of the original gender. That is why it changes from die Blume to das Blümchen (and uses the neuter article ein). Also, notice that the final -e in Blume is dropped before adding the suffix!

Why does Buch become Büchlein and not Buchlein?

When creating a diminutive with -chen or -lein, the main vowel of the word usually gets an umlaut if it is a, o, or u. So, the u in Buch becomes an ü. The same thing happens with Blume, which becomes Blümchen.

Why does this sentence use -lein for the book but -chen for the flower? Can I say Büchchen?

You technically could, but Büchchen is very hard to pronounce because of the two 'ch' sounds right next to each other. To make it sound better, German speakers prefer Büchlein. For most other words, like Blümchen, the -chen ending is much more common in modern standard German.

What is the function of um and zu in the second half of the sentence?