Start with LSTM or a
distilled version of a language model that is trained for generation task, e.g. GPT2.
There are a lot of pretrained language models (preferably at HuggingFace Hub). Take one, and fine-tune it on a much smaller dataset.
You can then pass prompts to let this model write poetry.
I would also suggest learning the fundamentals of Deep Learning, RNNs, LSTM, Transformer, etc. Because even if you know all these, this kind of taks is not trivial.
I would use HuggingFace + PyTorch for these kind of tasks.
Besides Deep Learning knowledge, you will also need to know your way around tokenizers, controlling parameters, evaluation metrics, etc.
Having some basic understanding of poetry is helpful, but to be able to truly apply your knowledge of running your poetry journal, you will really need to be on the proverbial edge. I hope you get there, and maybe get some papers out!
Although I worked on a language model because none existed in my language, my expertise (and employment) lies in Computer Vision. So, take what I said with a grain of salt.