Thanks for all the replies! A good variety of responses here; I tend to fall in the category of shoes always on when in public but I'm not that fussed at home.
I like the idea of getting her used to shoes and hat outside though; for those who have always had their kids in shoes from a young age, how did you deal with the tantrums and refusal, and also which shoes are the easiest to get on. I bought some pre walkers from Target but they're a nightmare to get on! It takes forever and I have to squash her feet into them which she hates, just not worth the effort.