But beyond that - what specific things should I be aiming for on a lab visit? I am still in the early stages of contacting the lab etc. But I think I am going to need to say more than I love your work... What can I get out of it? Or should I be thinking what can offer them?
As with any application, ideally you should highlight both - what they get out of it/why they should be offering you a place (your qualifications, skills, creativity, past accomplishments), and why you want to work at this specific place (your motivation).
As for finding more reasons to go to this specific lab than just learning new skills there - do they have any special equipment you could use? I knew someone who did a lab project in another country because she needed a special piece of equipment for her PhD research that was not available anywhere in her country (or maybe not for her field/purposes). Does this lab you are interested in have anything for you to play with that your home lab doesn't? ;)