Learn Russian: Честно говоря, я не читал эту книгу.

Говоря is an imperfective verbal adverb (sometimes called a gerund) meaning 'speaking'. The set phrase честно говоря literally translates to 'honestly speaking'. In Russian, introductory phrases like this one must be separated from the rest of the sentence by a comma.

Why is the imperfective verb не читал used here instead of the perfective не прочитал?

When denying that an action ever happened at all, Russian generally prefers the imperfective aspect to express a simple statement of fact. Using the perfective не прочитал would imply a failure to achieve a result, suggesting you were supposed to read it but didn't finish or didn't get around to it.

Since the verb is negated, why isn't the object in the genitive case (этой книги)?

You are thinking of the Genitive of Negation, which is a great concept to keep in mind! However, in modern conversational Russian, the accusative case is strongly preferred when the negated object is specific, concrete, and definite. The demonstrative pronoun эту makes the book very specific, so the accusative эту книгу is the most natural choice here.