To me, both sounds a bit clumsy. I would say this like this:
- 趣味は書くことと、素敵なノートや文房具を集めることと、テレビゲームをすることです。
- 趣味は書くことや、素敵なノートと文房具を集めることや、テレビゲームをすることです。
You can simply list two or more things ("A, B and C") using や or と. See other questions for the difference between the two, although the nuance would be small in this case:
If you used も here, the feeling of "also" would be emphasized ("not only A, but also B, and also C"), which I think is unnecessary.
If you used て here, the feeling of "and then" would be emphasized ("A, and then B, and then C"), which is not unnecessary, because these things are not relevant with one another. And "～て、～て、～も" is not grammatical. You have to drop the last も.
You can say, for example 「趣味は食べて寝ることです」「趣味は毎朝早く起きてランニングをすることです」, because the two actions are temporally connected.