Soy milk, rice milk, I even heard of someone making oat milk.
Go heavy on the fresh fruits and veggies, they are good for you anyway.
Use non-instant oatmeal, it really doesn't take that much longer if you use the microwave. Try other hot cereals (Maypo, cream of wheat, polenta, Ralston) for variety. There are 5-grain and 7-grain cereal mixes out there, they will take longer, but you can put them in the crockpot the night before and wake up to an already-made breakfast. Corn-meal mush is good too, some people like it with syrup, others with margarine (in your case dairy-free of course), salt and pepper.
Snacks--many crackers should be dairy-free (check those labels!!); peanut butter and other nut butters can go on toast, on crackers, or be spread on fruits and veggies. Fruit leather. Applesauce. Baked potato with salsa, or olive oil plus herbs, or non-dairy margarine, or pesto, or...I'm sure you'll think of more! Baked sweet potato. PB&J sandwich.
Check rice cakes to see if they have dairy, there are so many flavors you'll have to check each type (yep, a real pain--but you may find something).
Hummus--middle eastern food made of pureed garbanzo beans, olive oil and garlic. Dip veggies in it or spread on crackers or pita bread or toast, yummy! You can buy it already made or make your own.
Just thought of this--you could puree other types of beans (black, pinto, etc) for a slightly different spread.
Honey on toast is good too, or peanut butter and honey.
Watch your breads and store-bought soups, seems like a lot of them have dairy slipped in.
Some chocolate chips may be dairy free--Aldi's used to be but now they have something called "butteroil" which they say has milk :-(. With dairy-free chocolate chips you could make trail mix by adding raisins and nuts. Or do trail mixes without chocolate.
You may be able to tweak recipes calling for melted chocolate by substituting cocoa powder plus some fat. Most cookbooks should have this in the "Substitutions" section, if you don't find it email me and I will hunt it up for you.
Also, substitute crisco or your dairy-free margarine for recipes needing butter/margarine.
In my opinion, anything you eat for a meal you can eat for a snack, in smaller quantity (e.g. a cup of soup, leftover pasta). The opposite is also true--make a bunch of snacks into a meal.
If you have a health-food store nearby, visit them and see what they have.
Easy Peanut Butter Cooky Recipe:
1 cup peanut butter
1 cup sugar
splash of vanilla and/or dash of cinnamon if you want
Mix all. form into balls, put on greased cooky sheet or baking stone (ungreased). Criss-cross with fork if desired. BAke at 350 for 10 minutes? or until done. If you find dairy-free chocolate, you can put a chunk on top of cooky while baking, or right after you take cookies out of oven. This recipe is also gluten-free.
Sorry to run on. Hope this helps--a friend went dairy free and feels MUCH better, so it is worth it!