TL;DR: For cookies, use soft white wheat (or try mixing in spelt). And be aware that you may need to tweak some quantities of things like oil, baking powder, and sugar when you use different flours.
There are several types of modern wheat readily available for milling. The basic divisions are
- Red or white
- Soft or hard
- Winter or spring
Red means that the bran literally has a slightly redder color to it, but it is also a little tougher, with a slightly stronger flavor, and generally more noticeable in the finished product. Unless you're specifically going for that rustic wholemeal sort of effect, choose white wheat.
Soft and hard refer to how hard the berry is, but is also strongly correlated with how much gluten you get — soft meaning there's more starch and less protein, and hard being more protein and less starch. Basically, if you're baking bread you want hard; if you're baking cookies, muffins, cakes, etc., you want soft.
Spring wheat is (usually) planted in the spring and harvested before autumn. Winter wheat is (usually) planted later, so that it can't be harvested before winter; it goes dormant, and is harvested once it finishes growing in the spring. (In warmer places like parts of California, spring wheat actually grows during the warm winter, before the desperately hot summer arrives. It's still spring wheat, though, because of its genetic makeup.) Winter wheat tends to have slightly lower protein content, though the main driver is soft vs. hard.
Of course, in addition to the modern varieties, you'll also find ancient varieties like spelt, emmer, khorasan (a.k.a. kamut), and einkorn. These have slightly different flavor, and typically less gluten. The idea is that humans have bred wheat for millenia, starting from einkorn, to increase the gluten to provide more structure in their bread. But all that structure that's so important for bread is too much for more delicate baked goods like cakes, cookies, pancakes, etc. So you want to use one of the lower-gluten varieties for those. For example, you might try milling 50% soft white wheat and 50% spelt for those cookies, and I bet you'd get a great result.
All of these varieties still have bran, which is usually sifted out of all-purpose flour, pastry flour, cake flour, etc. When you use whole-grain flour (or mill it yourself and don't sift it), that bran tends to interrupt the formation of the gluten network, giving your baked goods a bit less structure — though maybe grittier, and possibly tougher or more bitter if you don't adjust your recipe.
That's not to mention rye, barley, and triticale (a hybrid of wheat and rye), which are the other gluten-containing grains. And then there are completely different grains like oat, buckwheat, millet, sorghum, teff, amaranth... (King Arthur Flour has a nice guide for this group here.)
All of this is basically saying that — even just among basic wheat varieties — there's a wide world of grains out there to try in your baking. Whether you buy them as flour, or mill them yourself, you'll really improve your skills if you start branching out by using new grains in your baking. But not all grains can be substituted for others. You need to think more, and maybe adapt recipes. Adjusting hydration levels is the first and most obvious thing. But sometimes you'll find that it helps to do things like adding some extra baking powder or adjusting the quantity of sugar to get your cookies just right.