Learn Russian: Я хочу увидеть море.

We use the perfective увидеть after хочу because the speaker wants to achieve a specific result: finally laying eyes on the sea. They are focusing on the completion of that goal as a single event, rather than the ongoing process of looking at it.

Could I use the imperfective and say Я хочу видеть море instead?

You could, but it changes the meaning to an ongoing state. You would use the imperfective видеть if you were asking for a hotel room with an ocean view, meaning you want the continuous ability to see the sea from your window.

Why doesn't the word море change its ending? Doesn't the verb to see require the accusative case?

It actually is in the accusative case! The verb увидеть is doing an action to the sea, making it the direct object. However, море is a neuter noun. In Russian, neuter nouns look exactly the same in the accusative case as they do in their dictionary form.

Why is only the first verb (хочу) conjugated, while увидеть stays in the infinitive?

When you use a verb of desire or ability (like wanting, needing, or being able to) followed by another action, only the first verb matches the subject. The second verb stays in the infinitive form, just like in English where we say I want TO see instead of I want I see.