Question

The other girls picked on her because of her size.
-----------------------------
Q1: Is 'picked' a transitive verb or an intransitive verb? Is 'on' a preposition or an adverb? What is the meaning of 'pick' here?
Q2: What other expressions can we use instead of 'picked on'?

Answer 1

It depends on how "pick" is used. In some uses, it is transitive, but in others it's intransitive.
http://www.merriam-webster.com/dictionary/pick

In this sentence, "on" is a preposition, with "her" as its object.

You can also use these expressions:
teased her about...
ribbed her about ...
joked about ...

Answer 2

The verb "to pick" means to pluck something, as one picks a flower, or picks a guitar string, or to pick something up off the floor. It may be a repeated action, as in picking at something that is bothersome. In the case of your sentence, it is an analogy for teasing someone or making fun of someone, usually repeatedly.

"on her" is a prepositional phrase used as an adverb to modify "picked". "On" is always a preposition. With its object, "her", it's a prepositional phrase, which may be an adjective or an adverb phrase. You might say the other girls teased her, or made fun of her.