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".

Both sentences mean pretty much the same thing, so you'd say: Prefiero hacer surf. (I prefer to surf/I'd prefer to be surfing.) It effectively means the same thing as both sentences. You could also say: Yo preferiría que yo estuviera haciendo surf. (I would prefer that I was surfing.) It's more convoluted, and means the exact same thing as the simpler version.

-Quiero hacer -Quiero estar haciendo (with infinitive) -Preferiría estar haciendo (with infinitive)

Cierto...