Is this sentence correct? "What I want to do is read this book."

"What I want to do is read this book."

Is it correct? Or, can I say:

"What I want to do is to read this book."
"What I want to do is reading this book."

Are all of the three sentences correct?

The infinitive read
-- or to read if the complementizer to is retained (it's optional) --
is correct, and the gerund reading is not correct.

This is what's called a Wh-Cleft sentence (also "Pseudo-Cleft"); it's a variety of Clefting, which is a syntactic device to emphasize certain parts of sentences. Clefts come from simple sentences like

• I want to read this book.

which they then proceed to dissect into parts, add various markers, and mix'n'match, viz.

• What I want (to do) is (to) read this book. (Wh-Cleft, or Pseudo-Cleft)
• What I want to read is this book.

• This book is what I want to read.

• All I want (to do) is (to) read this book. (All-Cleft)

• All I want to read is this book.
• This book is all I want to read.

• All I want is to read this book.

• It's this book (that) I want to read. (Cleft, or It-Cleft)

These are all grammatical, and they all mean the same thing, which is

• I want to read this book.

The reason why read and not reading is correct here is that want takes an infinitive complement, not a gerund. Do is an infinitive, so read has to be an infinitive, too.

If you use a cleft construction, you have to remember what the main verb is, and match it up correctly. Even when it's been moved away, the verb still governs what kind of complement it takes. That's one of the costs of complex constructions.