The verb has to agree with the noun. 'Books' is a plural noun, so you should use a plural verb: How many books are there? There are a lot of books. There are plenty of books. There are many books. You use a singular verb with uncountable nouns: How much information is there? There is a lot of information. There is lots of information. There is plenty of information.

the proper use of there is and there are has to do with whether or not the subject is singular or plural. There are is used for plural and there is used for singular. ex. There is a book - singular/ ex. There are some books - plural Using "plenty" is mostly the same rule.