I've always done the grocery shopping and I love it! I like cooking and eating so I see it as an extension of that rather than a weekly chore. It is much more enjoyable when the kids aren't with me though - one is OK, two is a no-no, hence I usually go when they're at school. If I'm not in a rush I quite like browsing the fruit & veg and health food sections...
We don't have coupons in Australia either, but the stores usually have specials of some kind and reward/loyalty schemes... I don't take much notice of any of that though because those deals don't usually apply to things I buy. I try to meal plan each week, buy what I need based on that + basics (milk, bread, yoghurt, eggs, fruit & veg). We eat alot of salads and veggies and I tend to buy mostly whole foods, things that are in season and no convenience products... this all keeps my grocery bill down.
I also grow some veggies at home, and I have a herb patch too... I'm far from self sufficient but it saves a bit, in summer at least.