Learn Russian: Мне нужен билет туда и обратно.

In Russian, to say you need a specific object, you use a structure that literally means "To me is necessary a [noun]." The person who needs it takes the Dative case (Мне). The thing that is needed is the grammatical subject in the Nominative case (билет). The word for "necessary" (нужен) must match the gender of that subject. Since билет is masculine, we use нужен. If you needed a visa (виза, feminine), it would be Мне нужна виза.

Why use туда и обратно for a return ticket? What does it literally mean?

Literally, туда и обратно means "there and back." This is the standard, everyday phrase Russians use for a round-trip or return ticket. If you just wanted a one-way ticket, you would simply ask for a билет туда.

Could I use там instead of туда here?

No, because Russian strictly separates location and direction. Там means "there" when talking about a fixed location, like where a building is. Туда means "there" when talking about direction or motion towards a place. Since a ticket is for traveling to a destination, you must use the directional word туда.

Is нужен a verb in this sentence?