I am not Alicorn, but I also like talking about delicious food and I do not eat eggs and dairy. Unfortunately, there is no general solution to the egg/dairy substitution problem, especially for the eggs end of it.

There are some things I just don't try to adapt: meringue, pastry cream, and whipped cream fall more-or-less into this category. I have had delicious dairy-free versions of whipped cream that seem to have been based on the fatty part of coconut milk, but I haven't made any myself.

There are some substitutions that are easy and consistent. In baking cakes, cookies, and similar things, you can usually use any unsweetened soy or nut milk 1:1 for milk, and use margarine in place of butter, or mild flavored vegetable oil in place of melted butter. It is easiest to get good results if your recipe is for spice or chocolate cake, or is otherwise meant to taste like something other than butter, as even the best non-dairy butter substitutes do not taste quite like the real thing. Eggs are a slightly harder thing to substitute for, so for a really easy experience, go for a recipe that does not use them; sometimes these are "light" cakes or recipes written when food was expensive or rationed.

Eggs, even in baking where they are non-obvious in the final product, can be tricky to substitute for because they do so many things. If the eggs are mainly adjusting the consistency of the batter or dough, you can substitute for 1 egg with 1/4 cup of soft silken tofu , applesauce, or soy yogurt, or anything of a similar texture that you think would taste good. If I expect the egg to actually do some work on helping the rising process, I use 1/4 cup of the liquid from the recipe or of soy milk, plus 1 Tbsp ground flaxseed or 1 tsp ground psyllium husk. If there are more than 1 or 2 eggs called for, I re-evaluate whether I want to use this recipe (things that are supposed to get flavor from eggs, or that use eggs in complicated ways, like with yolks and whites separated, are beyond my skill level to adapt), and if I still want to, I use some combination of the substitutions available to me, to avoid the food tasting heavily of flax or applesauce when I didn't intend that.