Sorry, I have doubts. ("I've" is only used in front of another verb.)
Which is the correct form to use with like?

Both forms can make correct sentences. The "like + to..." form is more general than the "like -ing" form. 

Usually, a person says "I like to read books" when she is describing her interests. "I like reading books" is usually used when the person is currently reading a book.

For example:

-I tell my boyfriend "I like spending time with you" when we go out to dinner.

-I tell acquaintances "I like to read and to spend time with my family."

  --I could also tell acquaintances "I like reading and spending time with my family" but the first form (like + to...) sounds better.

So, "like + to..." is better for speaking generally.

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If you use "would" in front of "like" (I would like...) the next verb must be in the infinitive form: I would like to read a book.