Gerund phrases always function as nouns.
Participial phrases are formed verbs but act like adjectives.
Independent phrases are groups of words that can stand on its own as a sentence. They have a subject and a verb.
Infinitive phrases are groups of words that use the infinitive form of a verb, they could work as a noun, adjectives or adverb.
I will put one example of each way that the infinitive phrases can be used.
Noun: I don´t want to study for my test.
Adjective: I need a magazine to read on the train.
Adverb: I went home to study for math.
I hope this answer helps you.