That's Andee on the left, Chickory on the right
Once housing is taken care of, that leaves your interaction with them. As a side note, it's always good to have a companion for them, either another youngster or an older horse. Until they are in the summer of their yearling year, most of what you do with them will be just getting them to lead nicely, pick up their feet, behave for your hoof care provider, stand tied, and tolerate grooming all over. If you plan on showing, you can add in training specific for the class you show in, eg. halter, showmanship, in-hand trail, etc. Since I don't show, I can't advise on what you can do for that.
As a yearling, you start introducing other things, such as trailer loading, ponying, sacking out.
Rio trailer loading at age 11 months
You also increase your expectations of good behaviour. If you plan to introduce them to riding in their 2 year old year, then in the fall of their yearling year you can introduce them to longeing for short periods (not too long because it's hard on their joints), sacking out with saddle blankets, and progressing to setting a light saddle on them.
Longeing Sassy as a long yearling
I like to have them checked by an equine dentist and get their wolf teeth pulled before they are 2. There is lots of ground work you can do with them as a yearling, like getting them to move away from the pressure of your hand on their side (imitating leg pressure). There are many books written on the subject of training, and I'm no expert, I just do what works for me.
None of this is rocket science, and there is no horse-whispering- mystery involved in getting a youngster prepared for their life as a good using horse. It's all common sense, and treating them like a horse, not tolerating bad behaviour just because they are cute, and laying out a training plan for them. If you aren't confident in being the first one on their back, plan ahead and save enough money to send them to a reputable trainer for at least a month. Remember, the ground work you do with them helps to build a bond and establishes your "place in the herd". Start as you mean to continue: if you don't want to be pushed around by a 1100 pound horse, don't let a 500 pound baby push you around. Establish boundaries from the get-go.
The end result will be a horse that you know the complete history on, one that you helped form, that you know inside out.
Me and Sundee, Coyote Belle's mama; my first heart horse that I bought as a weanling.




We covered about twenty miles that day, and she worked every minute of it and still was willing to give me more at the end of the day. That mare had heart!
The next year we hauled down there again; she loved working cattle and I could always count on her if we had to bust through trees and brush to head off a bunch quitter. I rode her in many parades, and won a little money team penning on her. I loved riding this mare, she was feather light in the face and leg, always willing, and she had no quit in her. 
She had the kindest disposition, all the other horses loved her- she fit into any herd she was introduced to.