1. The big boy picks on me at school.

2. The big boy harasses me at school.
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Q1: Are both the same in meaning?
Q2: What are the parts of speech of both 'picks' and 'on'?
Q3:picks on 1) an intransitive verb + a preposition? 2) a transitive verb + an adverb?
Q4:Do you use the expression 'picks on' frequently? What meaning does 'pick' have here?

1. They mean about the same.

2. 3. Picks is a verb. On is a preposition. However, in this sentence, picks on is used as a verb.
4. Yes. Picks on is a common expression. It means verbally and sometimes physically assaults.