Technological advances continue to redefine the dynamics of human-machine interactions, particularly in task execution. This proposal responds to the advancements in Generative AI by outlining a research plan that probes intent-AI interaction across a diverse set of tasks: fixed-scope content curation task, atomic creative tasks, and complex and interdependent tasks. This exploration aims to inform and contribute to the development of Intent-based User Interface (IUI). The study is structured in three phases: examining fixed-scope tasks through news headline generation, exploring atomic creative tasks via analogy generation, and delving into complex tasks through exploratory visual data analysis. Future work will focus on improving IUIs to better provide suggestions to encourage experienced users to express broad and exploratory intents, and detailed and structured guidance for novice users to iterate on analysis intents for high quality outputs.