Cross contamination is definitely the big issue here. When possible, don't even bake the GF cakes on the same day you bake a regular cake. Flour dust can hang in the air for several hours after you've used it, and even that is enough to make someone sick. Use an entirely separate set of equipment. If you can't do that, wash them really really really well. As you're testing, have a brave GF person try them if you can, and be aware that symptoms don't usually show until at least several hours after they've eaten it, or even a day. This makes it hard to know exactly what they ate that was the problem.
It might just be easier to doctor some mixes. Amy's is always good but my favorite is the Namaste brand. Be aware they don't go as far as a regular mix so you will need more of them, and they're far more expensive. Everything GF is more expensive. For flours, follow the rule of 3's and use 3 different flours, not just one (for example, sweet white rice flour, almond flour, and coconut flour). Every single ingredient must be GF, including your vanilla, so watch out for that.
Both books I saw recommended above are good, but as someone else said, it takes a loooong time to get GF baking right. It's also much more susceptible to changes in temparture, humidity, etc, than regular cakes. You won't be able to carve a GF cake or put it through a lot of stress and stacking, it just won't hold together.
I've been GF myself for 4 years and I offer regular cakes and GF. I have to wear a mask whenever I start to use regular flour. Good luck!