If by immortal you mean `not being able to die, like Captain Jack Harkness for instance´, I´d just go fighting evil or something. perhaps take a nice swim across the pacific or something. after that, I´d probably just go for a walk and take pictures of where I´ve been. say, sahara, Russia on barefoot, China, America, London. Stick around where it´s fun, go away from where it´s not.
for the people unfamiliar with Captain Jack Harkness, he´s a character from the new Doctor Who, who´s turned invincible by mistake (the original idea was to bring him back to life, but it was a tad overdone[just a tad])