If you are thinking about trying to cut dairy from your diet, there are a lot of vegan friendly ingredients that make it pretty easy. Below are some substitutes that you could try.
Rice milk and soy milk are the most common replacements for cow's milk. Both of these are readily available in most stores. Ricedream rice milk is sold at Super Walmart back in the refrigerated dairy section. It's also available in the non-refrigerated boxes (but I don't think it tastes quite as good). In my opinion, rice milk tastes the most like real milk and it's white, so it looks about like skim milk. Even if you aren't ready to pour it on your cereal, it's great in cooking (substitute 1 for 1 in any recipe). I think pretty much every grocery store sells soy milk now, both refrigerated and non-refrigerated. Soy milk is creamier in texture and has a slightly stronger taste. It also is more of a cream color (I've been told by some that it's harder to switch when it doesn't look like milk). There are several different brands and they all taste slightly different, so you can just experiment and see what you like. Soy milk (especially Silk's Very Vanilla Soy) works great in smoothies and cooking because it's thick and creamy. Although I haven't found a great rice milk yogurt, soy yogurt is available and I think it's pretty good. Chocolate soy milk is also a treat you might want to try.
The best egg replacer that we have found is Ener-G Egg Replacer. It is about 5 dollars a box and you can buy it at most health food stores (as well as Kroger if you live in TX). One box lasts me about 6 months, so it's actually quite a bit cheaper than eggs. While you can't make scrambled eggs or omelets out of it, I've never had a problem replacing eggs in baking. I just follow the directions on the box and it works great. You can also use flaxseed ground up in a blender with water or milk but I typically just go for the easier route and use Egg replacer.
Cheese is a little harder to replace unless you just use fake soy or rice cheese. These are high in fat and highly processed, but a good substitute if you need cheese on something. Some melt well and some don't (below is a picture of one of my favorite melters). Although I thought I would really miss cheese, I actually don't miss it at all. We use avocados a lot as a topping, and I think it's a great alternative. Nutritional yeast can also be used to make things taste cheesy. My kids actually call nutritional yeast cheese and like to eat it plain (?).
A lot of people can't stand the thought of no icecream, but there really are some great substitutes. You can get Ricedream Icecream and Soydream Icecream in a ton of different flavors. There is also a new Coconut Milk Icecream that I really want to try. While most of these are still quite high in fat, they serve the purpose of a rich, creamy dessert.