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If your "dairy" concern is butter, it can successfully be replaced with margarine in many cases. Coconut oil will also work, but can require additional recipe adjustments.
If you are going to use margarine, read the label carefully to be sure it is dairy-free. Most contain "whey" which is a dairy product. I do quite a bit of dairy-free baking, and use Saffola as it is one of the few whey-free margarines that I've found. You can also use Crisco (or even butter flavored Crisco) but I'm not particularly fond of either product.
So, whatever it is you'd like to make (from chocolate chip cookies to cakes) try margarine instead of butter.
If your concern is for a milk replacement (e.g., some cakes require milk) try rice milk or (for a richer result) coconut milk -- one for one. Many of these alternative milks also have some vanilla flavor added, so just be aware of that in your purchase (i.e., choose it if you want -- and you may need to slightly adjust the vanilla in the recipe; or choose the without). I've successfully made some lovely custards using coconut milk instead of cow's milk (interestingly, you can taste the coconut before the item is cooked, but rarely after -- the coconut flavor tends to disappear in cooking.)
I've heard very good things about some of the nut milks (like almond milk) as a replacement, but we're a nut-free household (severe allergies) so I don't have any experience with them.
Back to butter -- just know that if you use a margarine replacement, you will almost always get a good result but it won't be quite as rich. As an example, I've "successfully" made lemon curd many times with margarine instead of butter (looks good; texture good; good flavor) but it just isn't as rich as a butter based lemon curd.
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