Let me ask this about your examples, NileCat:
--->本を読みはじめる (Using a verb stem here, you just put the verb you want right after. Is this meant to be a correct example or incorrect? I couldn't really tell based on what you wrote)
--->本を読むのをはじめる (This appears to be what my book says. Is this also correct? What is the best and most natural form of all of these?)
At this stage, as NileCat-san says, it is more useful to understand the grammar ～Vはじめる／～Vだす rather than V+[こと／の］をはじめる to you.
Both ～Vはじめる and ～Vだす, ます form of V is used, are categorized as JLPT 3 grammar.
The girl began [to cry/crying].
The girl stated to cry.
-- ～V(ますform )+はじめる
-- 泣く→ 泣き(ますform)＋はじめる＝泣きはじめる
-- past tense →泣きはじめ（まし）た
-- past tense → 泣きだし（まし）た
Let's review NileCat-san's examples.
"その女の子は泣く［の／こと］をはじめ（まし）た" is hardly used in a daily conversation. (When the verb is neither an intentional nor a conscious action, V ことをはじめる is an inappropriate usage.)
An inanimate subject works as well in both usage.
The train began [to move/ moving].
The train started moving.
If the subject of a sentence is inanimate, Vことをはじめました is weird in most cases unless it has a poetic/rhetorical intention. At least その電車は動くことをはじめた won't be replaced with その電車は動きはじめた.
In either case, grammar Ｖはじめる/Ｖだす is much more common and has a natural tone than Vことをはじめる.
As for your original, 本を読みはじめる is natural, whereas 本を読むのをはじめる is uncommon.
Hope this helps.
Edit: It seemed that I had taken too long to write this post.
Now I'm guessing your book might say 本を読むのをやめた
. ( not 本を読むのをはじめた