I'm a native speaker, and I don't know all of the grammatical terms you're using. There's a small set of basic grammatical terms that are common knowledge among (educated) native speakers. The very refined terminology of linguists includes many terms that are not common knowledge, and uses some terms in ways that conflict with common usage. That is, even a college-educated person or a professional writer is unlikely to know many of the grammatical terms in your question.
You should know that it is not necessary to know sophisticated linguistic jargon to learn English. Learning a language is not learning facts about the language, but learning the skill of communicating through the language, which is mostly subconscious.
I'll explain some of the grammar of your first example sentence using the common-knowledge terminology. (If you want help with another sentence, it's best to ask a separate question.)
I don't want
Do functions as an auxiliary verb here. It adds nothing to the meaning. It's only there to help negate "want". (Negation is a big topic, so I'll stop here.) I would not call it a modal verb. Modal verbs add to the meaning: they add things like tense, obligation, or estimate of probability.
This announces that what the speaker doesn't want is about to be expressed by a verb in the infinitive:
"Go on" is a phrasal verb--in the infinitive, as promised. Its meaning here is the same as "continue".
"Knowing" is not an infinitive. Most people would say it's a gerund here, but I think one could also argue that it's a present participle. A gerund is a verb in its -ing form, which functions as a noun within the sentence, but also can command its own verb clause with objects and even a subject of its own. In this sentence, "knowing" doesn't have a subject of its own, but it does have an object: "you".
So, "you" is the object of "knowing". "Knowing you" is the object of "go on". "To go on knowing you" is the object of "want". "Want to go on knowing you" is negated by "don't". The subject of the whole thing is "I".
Does that help?
Note that people can reasonably analyze the sentence in different ways. English grammar is not crisp and clean. English sentences are filled with little phrases that sometimes overlap to reinforce the intended meaning. For example, one could treat "want to" as a familiar phrase, and one could call "to go on" an infinitive. They overlap at "to", so it doesn't really matter. What matters is that a fluent speaker, who is accustomed to the many ways these words and phrases can combine, can understand how all the various parts of the sentence fit together.