Since the question can be answered with mostly opinion based statements, I hope you would forgive me if I could not cite halakhah for all my claims. I dare to give an answer due to my (limited) experience as a chazan.
First, the purpose of different piyutim should be considered. Our sages of blessed memory composed these religious poems in order to increase the solemnity of the service and on yomim noraim to induce introspection and self-reflection. The piety of authors, such as R' Eleazar haKalir or the Chasidei Ashkenaz should serve as an example of right attitude on these days. Due to the complexity and often metaphoric meaning, one should do efforts to read and understand these poems in advance. Many of them feature Maasei Merkavah, which might be difficult for advanced learners as well.
Therefore one should really know what can be considered obligatory (mostly based on minhag), and what is an additional feature that is a very meritorious act to say. In Ashkenazi practice piyutim are inserted between the blessings of the Shema (yotzer, ofan, zalut) and in the repetition of the Amidah, mostly around the first two blessings and between the verses of the Kedushah.
Although I'm not aware of clear halakhic reference, but all of them are optional in theory. Some even oppose such additions, this is why we ask permission in advance to say them (Misod chachamim), but Mizrachim barely add piyutim to the repetition. Orach Chayim 620:1 praises the practice of limiting piyutim in the morning prayer of Yom Kippur to finish musaf on time.
The selection among these poems are more treacherous waters, and based on the local minhag the rabbi and the chazan should agree on the parts to be said. A very common choice to omit are the selichot of Yom Kippur. Others may be selected on the indications of the machzor. My Schlesinger books rarely say that certain parts are omitted by many congregations, but very fine print and the lack of translation might give a clue. Artscroll took one step ahead and, after consulting local rabbis, put many piyutim in appendix. To the contrary, my teacher always said as a rule of thumb that those poems shouldn't be omitted, when the Ark is opened.
So there is a clear obligation only for the Shema, its standard blessings and the parts said in the silent Amidah, but I could hardly imagine a Yom Kippur service without Untaneh Tokef or the service in the Temple to name a few. Everything depends on the local minhag, which should reflect the frumkeit of the congregation and should avoid tircha detzibura.
I suppose Artscroll has good indications to start with, and you could focus on the parts that serve your spiritual growth at the current level. Since it's very hard to maintain a proper attention even on a short weekday service (see discussions about Pitum haketoret and the Akeidah), I would suggest you to start with less, and proceed if you feel familiar with the recited parts (see also Orach Chayim 1:4).
Suggested sources are more than welcome.