To surf is "hacer surf" or even "hacer surfing" (which sounds more natural to me). Also "practicar surf" or "practicar surfing" would do.
The difference between "I want to surf" and "I want to be surfing" is one between a normal infinitive and a progressive infinitive, which is pretty much the same in Spanish. So in Spanish you can also say "Quiero hacer surfing" and "quiero estar haciendo surfing".
The expression "would rather" is usually translated into Spanish with the verb "preferir", either in the present or the conditional tense. So "I'd rather surf" would be "preferiría hacer surf" or "prefiero hacer surf". When you say "I'd rather be surfing" you're using a progressive infinitive and you can also reflect that in Spanish: "Preferiría estar haciendo surf" or "prefiero estar haciendo surf".