Learn Russian: Я хочу заняться спортом.

The verb заняться (to take up / get busy with) and its imperfective pair заниматься always require the object to be in the Instrumental case. You are essentially saying that you want to occupy yourself with sport.

Why use the perfective verb заняться instead of the imperfective заниматься?

The perfective заняться focuses on the start of the action, meaning to take up or to get into a new hobby. If you said Я хочу заниматься спортом, it would mean I want to exercise regularly without the specific emphasis on starting it as a new phase.

Could I use a phase verb like начать to say this?

Yes, you could say Я хочу начать заниматься спортом (I want to start doing sports). However, заняться спортом is a very natural and concise way to express taking up a new activity without needing to stack three verbs together.

Does заняться work like the phase verbs начаться or закончиться from this lesson?

No, it acts differently. Phase verbs like начаться (to begin) are intransitive and describe an event happening on its own with a Nominative subject, such as Матч начался (The match began). Заняться describes a deliberate action by a person (Я хочу заняться) and requires an Instrumental object.