Learn Russian: Я хочу изучать всю вселенную.

Всю is the feminine Accusative form of весь (all or whole). We use the feminine form because вселенную (universe) is a feminine noun. We use the Accusative case because it is the direct object receiving the action of the verb изучать (to study).

The dictionary word for universe is вселенная. Why does it have an adjective ending (-ую) here?

Great catch! Вселенная is a noun, but it originally comes from an adjective. Because of this, it changes its endings exactly like a standard feminine adjective. To put it in the Accusative case, the Nominative ending -ая changes to -ую.

Why use изучать for 'to study' instead of учить?

Russian has a few different verbs for learning and studying. Изучать is used when you are researching or deeply studying a complex subject or field (like a science, history, or the universe). Учить is mostly used for memorizing specific things, like vocabulary words, rules, or a poem.

Why is the imperfective verb изучать used here? Could I use the perfective изучить?

We use the imperfective изучать because studying the universe is a massive, ongoing, and continuous process. Using the perfective изучить would imply a completed result: that you plan to finish the action and learn absolutely everything there is to know about the universe, which is impossible!