Goat yogurt won't be as thick as homemade cow yogurt. Some people add powdered goat milk to make it thicker and creamier, or they add tapioca starch. I don't do grains so I haven't tried that but most store-bought goat yogurt has tapioca in it as a thickening agent. I have a batch of goat yogurt going now that I added a little gelatin to, which sounds weird but it worked so well with my coconut yogurt I thought I'd give it a try. It does not come out like jell-o at all, just like regular creamy yogurt. I haven't done it with goat before but I'll let you know what happens. I also ferment for 24 hours to make it thicker and more tart, with more beneficial bacteria. After 24 hours most of the lactose should be gone as well.
I always used to use an insulated cooler to make yogurt but I finally bought a yogurt maker that holds 3 quart-sized mason jars. It has saved me a lot of hassle, because if you do 24-hour ferments it is difficult to maintain the proper temperature without baby-sitting the yogurt all day, and you can still use your oven. It seems to keep the milk at a steady 100-102 degrees the whole time. It was not very expensive either.
One more thing, if you're using store-bought yogurt as a starter it could take longer to ferment than if you're using an actual starter or real home-made yogurt, since it doesn't contain as many bacteria to start with.