Answer: The sentence that is correctly punctuated is "I know exactly where to start my research: the library".
Step-by-step explanation: Punctuation rules state that a colon can be used to introduce an item or a list. Moreover, when a colon is used to introduce a second sentence, this second sentence must be closely related to the first one and illustrate it. Furthermore, the first letter of the word coming after the colon should not be capitalized. On the other hand, in general, a semicolon is only used to separate two ideas. In that way, the only sentence among the options that follows the punctuation rules is "I know exactly where to start my research: the library" since the colon is not followed by a capital letter and it has been used to introduce an item. In contrast, in "The library is a great place: I like to research", the colon is followed by a capital letter and by a second sentence that is not closely related to the first one, while in "I choose to do my research; at the library" and "I started my research; at the library" the semicolon is not separating two ideas, but separating a phrase from the rest of the sentence, which is incorrect.