The first thing you will need to figure out is an alternative to milk, especially if you are a fan of breakfast cereal. Soya milk is a good option to start with, but more and more dairy-free milks are appearing on the supermarket shelves. See Milk Alternatives for some other types of plant-based milk. The barista-style oat milk is a great in coffee and tea, or alternatively why not try other teas that don't need milk such as green tea?
To replace butter on your toast and in sandwiches there are a few dairy-free options. My favourite is Pure margarine.
Thanks to the popularity of plant-based foods, in recent years there has been a huge improvement in vegan cheese. They are normally coconut based and can be found in supermarkets in block form, sliced, or grated. I have to say I still don't think there is a perfect equivalent to blocks of cheeses such as cheddar, but there are some great vegan mozzarella-type cheeses that melt really well, and slices of what I call 'plastic burger cheese' that are just as good in a burger as the dairy ones.
I spend a lot of time looking at labels in the supermarket. It's a bit daunting at first but you will soon get to know which foods are safe. The good news is that since milk and eggs are common allergens you will see them in bold in the ingredients list. Milk can often be found in places you wouldn't expect. Here are some to watch out for:
Egg can also show up as a sneaky ingredient.
Pills sometimes contain lactose, so make sure to read the label. Look for capsules, for example paracetamol capsules with the plastic-type coating may not contain lactose whereas the traditional 'chalky' pills do. For some medicines, such as antihistamines there are liquid alternatives. Ask your pharmacist for advice.
If you cook from scratch you are on to a winner. After all, most food is dairy-free to start with. See the Recipes section for some ideas.