An example of a sentence including the verb “begin” in the present perfect tense would be : I have begun to study French. Meaning I started studying French at some point in the past and It has effects on the present.
An example of a sentence including the verb “begin” in the past perfect tense would be: She had begun to read well when she was 5 years old. Meaning when she was 5 she could already read properly.
An example of a sentence including the verb “begin” in the future perfect tense would be: By the time you arrive at the cinema, the movie will have begun. Meaning you will miss the beginning of the movie because you will be late.