So you've decided your horse needs some equipment? If you haven't purchased it already, you'll need to go to the Tack Shop. To edit the equipment on a horse, go to their page and find the link on the side to go to their Equipment page. From there, you can assign equipment you have already purchased. Equipment does wear out periodically, so be prepared to change it when it wears out.
When it has worn out, you'll see a red box with !! in it next to the equipment link on a horse's page. Be sure to double check each meet to make sure your horses are running with everything they need. You will also get a pop-up notice if you try to enter a horse who has broken equipment.
You are encouraged NOT to put a bunch of equipment on your horse for their first race, as this can make equipment notes very confusing. However, most horses will prefer running with a whip, so that is one thing you should go ahead and put on.
Information about each type of equipment can be viewed in the Tack Shop, which should help you determine what your horse might need.