Yes, in terms of household items, bring your appliances. We cleaned our vaccum (dyson) and got it through customs. I wouldn't buy one in UK and ship it though - you have to change the plugs anyway and you won't get your warranty out of it. I am glad I bought my tumble dryer.
Again, I wouldn't buy clothes etc. You never know what will be in fashion or fit in a few years - particularly with kids. If you really like a few brands, they can be shipped over for a reasonable price. Much better to do that than waste money on items that won't get used. Ditto curtains and household furnishings. I would brings quilts, bedding etc as they are expensive here - but you have to make sure you have the same size beds!
If you have pets, perhaps stock up on flea treatment - a friend of mine still orders hers from the UK!