I may not be the best person to respond, as I've been very allergic to dairy since birth. But I became vegetarian for ethical reasons 20 years ago and so had to learn to stop eating meat, which I had always found quite delicious and was part of nearly every meal until then. So I do know about how to change eating habits.
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It will be much easier to tackle the in-house dairy consumption before dealing with what everyone eats out of the house. And you can always work with one dairy product at a time or one meal at a time. I'd recommend looking at some vegan cookbooks (especially for baked goods/pancakes/creamy sauces/etc.) to get some dairy free recipes. It does take a bit of time for taste buds to adjust so even if one particular alternative doesn't cut it immediately, it may "pass" a year later.
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Milk - it is really, really easy to substitute alternative milks in baked goods. Typically you'd want to stick to a nut/hemp/soy milk as rice or potato don't always do so well. For breakfast cereals you could have your family have a taste-test to see if there is a favorite non-dairy milk or if none immediately appeal, then you could always blend a banana or other fruit into the milk and pour that over the breakfast cereal.
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Butter - for vegetable cooking, you could switch to coconut oil or olive oil. For spreading on bread, coconut butter is delicious. There is always margarine - just avoid the trans fat ones.
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Ice Creams - there are some delicious alternatives in grocery stores now. They are often expensive though. I have a friend who makes her own vegan ice creams and they are just amazing! Much cheaper too.
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Cheese - hmmm, this always seems to be the hardest for people. Maybe other posters will have suggestions there.
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And since you are planning to do this over an extended amount of time, you can make this pleasant for your family. Just keep trying out new things, keep going with what the family likes, and maybe modify the things that they aren't wild about to see if they like it in a revised form.
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Good luck!