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Taxonomies play a crucial role in organizing knowledge for various natural language processing tasks. Recent advancements in large language models (LLMs) have opened new avenues for automating taxonomy-related tasks with greater accuracy. In this paper, we explore the potential of contemporary LLMs in learning, evaluating and predicting taxonomic relations across multiple lexical semantic tasks. We propose novel method for creation of taxonomy-based instruction datasets. With the use of this dataset based on WordNet we build TaxoLLaMA, a unified model fine-tuned designed to handle a wide range of taxonomy-related tasks such as taxonomy construction and enrichment, hypernym discovery, and lexical entailment. The experimental results demonstrate that our model achieves state-of-the-art performance on 11 out of 16 tasks and ranked second on 4 other tasks. We also explore LLM ability for constructed taxonomies graph refinement and present comprehensive ablation study and thorough error analysis supported by both manual and automated techniques.