I'm from New Zealand, I don't know about sexist but it's definitely racist. (Have to point out I'm not saying the UK isn't either. Think UK is v bad for racism too.)
I think you have to expect that if you go anywhere that's ex-colony as there are land rights issues all over the place, and constructive prejudice programmes which are working but very unpopular. I don't think it will be a problem for ever, but it is definitely still a problem now. It would depend with OZ and NZ as well as anywhere else on where you went in the country as to how bad it would be. So if you wanted to avoid that you would have to do a lot of research to find out where the levels are acceptable for you.
Just about emigrating generally, do you have any pre-existing family or friends even in Oz? When we came to the UK from NZ I found it very lonely (as a child) because I missed out on all of the family things others were doing around me. Also find that in NZ (not sure about Oz) more people stay close to home as they grow up so it's quite noticeable if you don't have any, and groups can be very clique. Just from personal experience I would say NZ more friendly that Oz, but I know more people there, so I would, wouldn't I??
As far as the fit in or fuck off attitude goes I have definitely had a hellofa lot of that here in the UK. As a kid it was horribly confusing. It is not a sign of lovely welcoming multiculturalism, but frankly of narrow minded arseholes. If you're going to move to another country IMO you have to be committed enough to that new country to want to defend it should war break out, that doesn't mean you should also automatically cut all ties with your past or stop supporting your original countries sports teams. Many people will expect you to do this, wherever you move. This can be very tough, especially if you are homesick. As those signs of your past are very comforting, but you will get the piss taken out of you for wanting/needing them.
Would also add... you should really spend more than one trip there first! My Grandpa moved his family to NZ from the UK 40 years ago without ever having been there before, and it wasn't exactly what he expected. I'm going to assume you're probably not that daft(!)
TBH, if you have kids, unless you really hugely have a problem with the UK I wouldn't consider moving abroad. Even somewhere that speaks the same language and is also Americanised will be a massive culture shock, and unless you are a very strong and close family unit you will find it really tough to start with, and you'll have to seriously have the balls to stick it out.
Now I have lived in the UK for nearly 20 years, I'm unlikely to leave as have settled in, have lots of ties here. But when I was a child I was absolutely desperate to leave as felt so unwelcome. I don't think this is exactly isolated to the UK. It is hard to move 13,000 miles away and you should make sure your kids have a realistic idea of what it will be like.
But... that's only my opinion.