At 15 months he won't understand "time out," but he is plenty old enough to understand "no." Kids who whine do so because whining has worked for them. Choose which situations you are going to say "no" in, and then stick with it, because if you tell him no, and then give in after he whines and fusses, I guarantee he will continue to whine.
You better get the tantrum business out of the way before your next child comes, too, or you'll really be at your wits' end. The only reason a child has tantrums is that the tantrum has gotten them what they want. If you give him what he wants after he has a tantrum, he will continue to have tantrums. It's really that simple.
I would start with "no, you can't have a snack before dinner." You don't have to explain to him that snacking will ruin his appetite; he won't be able to understand that anyway. You don't have to explain anything, except that "mommy is in charge here." That's about all they really understand at this age.
For other situations, where you don't really want to get into it with him, you can often distract him from what he's whining about by physically swooping him up and redirecting his attention to something else: "LOOK, Honey! Here's your firetruck! How does it sound?"