Hi A.,
My son is allergic to dairy, too. A nice substitute for cow's milk and cow's milk products are goat's milk and goat's milk products. They make goat yogurt, cheese, butter, even ice cream! There are several milk "substitutes" on the market now: soy, oat, almond, and hemp. There are also dairy-free versions of cheeses, butter, and sour cream--usually these are soy or nut based.
Your absolute best bet is to find a food co-op or natural food store in your area. You will be amazed at how many milk-free products are out there. Regular grocery stores, especially if you live in a rural area, can be a joke for food allergies. You may have to plan a once-a-month trip into one of the larger cities to find what you need.
We don't eat out much and I make a lot of things from scratch now, like bread and crackers, because milk and milk-derived products are in everything. My son got hives from some cookies that had "milk powder" in them--you just have to be diligent about reading ingredient labels.
I can't stand the taste of goat's milk, so when I cook at home, I will often split a recipe (like a hotdish) and make half with cow's milk and half with goat's. Sometimes, we do eat stuff that my son can't have, and then he gets a substitute. For example, when I make quiche with cream, I will give my son scrambled eggs instead. We just had bean burritos and I left off cheese and sour cream on his. I feel like my son needs to get used to the fact that sometimes he won't be able to eat what other people are eating, and has to be OK with eating "substitutes" or foregoing certain dishes.
Once you get in the swing of things, managing a dairy allergy becomes second nature. Rice should be very easy to avoid--it's hardly in anything in this country, and when it is, it's very obvious. Just keep reading the labels. Let me know if you need more direction or resources.