The way I understand it, the -て form of verbs is used for many different things - very often, to connect them to other verbs or sentences. Not saying it's the same, but we could compare it to the English -ing: if I were learning English now, there wouldn't be much point in trying to memorise that I can use it to say "beginning to walk", "singing a song", "I don't like waiting". Or maybe I understood you wrong - what do you mean by "hiragana phrases"?
I think every て pattern that look like a set (てください、てはいけない、ている、ていく、ておく、etc) And the answer to that is too many, because they are not really sets, bundles.
For a Japanese, it's just て + something else. Meaning: when you will know more words, expression, etc, you will be able to combine them. You don't to memorize them (except a few at the beginning). Later, it's more like combining て with something else rather remembering phrases.
