Pictures would certainly help us help you.
If you can get a "mental" layout of "where" you want what, you can go from there. Simply looking at where that carpenter placed the studs, wall bracers, windows/shutters, etc will help you make use of those to create what you want inside.
The milking area will need a place to house whatever you'll be feeding the goats as you milk them (We use metal barrels with lids to keep mice out.), shelves for your milking tools (udder wash, etc.) & the milking stand. (I put a wooden floor in part of my milking area and at one end, I placed a 2' high wall just the length of a goat. That area is what I use for milking because, when I'm training a doe who isn't that anxious to be milked, she can't go anywhere as that wall stops her.)
Your hay storage area will be handy if there is a way to get the hay into their feeder bins from where the hay is. We have the main feeder bin (with its own step) on the other side of a wall from where we place the hay; and the top of that wall is not there; so all I need do is place slabs over the wall directly into the feeders.
As for baby goats, we let them stay with the dams; however, if you're going to separate them at birth, others can give better advice than I. [We did establish a small pen inside the barn strictly for special occasions. One would be to place a dam and her kids
if that dam wasn't wanting to bond as this forced the dam to bond with her kids. This special area is a solid structure but with strong wire so whatever goat(s) are placed in there for a time can still see the rest of the herd.] We also use this little pen in case we need to separate a sick/hurt goat from the herd.
One other thing I found quite useful is to design the partitions wanted inside the barn to fit different sizes of cow/hog paneling so I can separate the barn if need be. I keep those panels hung on the inside of the barn ready to use if need be. Oh I also divided up our barn in such a way that would contain a buck in rut if need be. It takes a very solid wall to do this high enough so neither the does nor the buck can jump over. This works beautifully throughout a bad winter when I house the bucks with the does. Then when the buck is not there, I simply open the solid gate, solidify it to the existing wall it is connected to and the entire barn is open for the does and their kids.
When you're designing, remember some goats are jumpers; even some kids do a good job of getting over fencing/walls; so make your partitions high enough to prevent this. I've even had grown does crawl under a fence that was only 18" off the ground!
As for a cow, I personally would not let a cow sleep where my goats sleep because their offcasts is not in pellet form is as that of goats; and I would not want my goats laying down their nice udders in such a curd.
Lastly, you'll need to plan on "hanging" water buckets and mineral/baking soda feeders. These need to stay off the floor to maintain cleanliness.
Here is a collage of what I designed that will give you some idea of what I've been describing. This has worked nicely for our small herd.
Good luck. I found designing our barn was a lot of fun and I hope your enjoy doing yours.