I did this with my son until he weaned at 16 months (started at one month). It turned out he had a dairy sensitivity, as I did (he outgrew his around age 3, no one can tell me when I outgrew mine). It was hard, but worth it for me. He was much less fussy (and stopped having projectile diarrhea--I kid you not). My husband wasn't so sure as it meant the food we ate together changed.
I used soy milk and pretty much everything else I just didn't bother with the soy, rice, or other substitute version of because I didn't care for them. It's just important to read labels carefully because some soy cheese, margarine, and other surprising foods have dairy in them. I used olive or canola oil to replace butter, occasionally used soy yogurt or sour cream in baking, but wouldn't personally eat them plain. I found some vegan recipes online that provide substitutes for dairy, but mostly I cooked things that didn't have dairy products in them. I can assure I totally pigged out on cheese, yogurt, and ice cream after my son weaned!
My second born had some fussiness early on and I was afraid we were facing the same thing, but it turned out to be me producing too much milk and reflux (which both kids and my husband have). I managed to slow down the milk (check out KellyMom.com if you have questions on that) and we got her on Zantac. But, she also has turned out to simply be more irritable and expressive than her brother. Good luck.