Maybe this was answered and I missed it elsewhere. Has cow been recently tested for TB? I would want the test done to make sure cow is safe to mik. ESPECIALLY using her milk for my children's consumption. I would consider a TB test essential, as part of her "cow expenses" you can't avoid.
You will want to get a place to tie the calf when you are milking the cow. She would probably be happiest if she can see him while in the headgate. I would think within a couple days of learning the routine, she would stand quieter, not be swinging about during the milking process. Milking time should be QUIET, relaxing so she lets down all her milk for you. Having goats underfoot, kids running about doing kid stuff, will make her worry, not relax for you.
And in my dairy experience, even with the calf at hand, you ALWAYS want to empty her when you start milking. Leaving ANY milk in there is not a good choice and can cause problems. Empty, stripped out udder, HELPS cow produce better and more quantity of milk.
If she is already trained for hand milking, you don't want her UNTRAINED and learning bad habits for milking time. So things need to be set up for easy milking, safe for you and her during the milking time. Not allow her to act poorly and GET AWAY with it. Cows learn fast, both good and BAD things that work to get her own way!!
Not sure if your headgate has sides or is just the locking device. If you have a framework, you may want to put a board (solid) across the to sides and fasten it to prevent her rear end sashaying about during milking. High enough to hold her in place, and allow you to reach under and milk comfortably. A piece of older carpet or rubber mat as a floor in the chute area will keep dirt and dust down, give her better footing, so milk is not getting dirt settled on it.
Begin as you mean to go on, set things up so you can milk her with everything under control, allow you to control of her situation surroundings. Once a cow starts arguing, fighting with you during milking, things go down hill and it doesn't end well. Remove her worry about calf by having it visible and close, no NOISY other animals or children running about while milking is going on. YOU get upset, she can feel that and gets upset as well.
Not sure of your dairy experience, but sounds like you need to read up on it, learn WHY your management of her udder is going to help or make problems for her. Clean handling techniques of fresh milk from the cow to the table must be observed to keep milk safe.. My Gramma was EXTREMELY particular on how the milk was managed before it got to the drinking stage. Straining, stove top pasturizing with thermometer and prompt chilling after, were top priority with her. Dirt was almost a swear word in relation to her cow's milk! But those were the kind of practices needed to keep milk safe in those days for people to drink.
we're all so excited! Especially the kids, as they're really lookin forward to fresh milk!!! 
