Hey there Lyssa, congrats on breeding some of your first foals!   After a foal is born it will end up in your transport, you will want to go ahead and either board it at home or someone elses boarding center and then go take care of it (feed and water).   After this you can begin foal training!  While basic care can be found on the first horse tab (the circle i), training is found on the barbell tab.  Foals have 7 stages of foal training and have until they reach 2 to complete it.   Once they turn 2 foal training will disappear if uncompleted.   You can find the wiki page for foal training here [Wiki Page]

Once a foal finishes foal training, as long as they have a disc and/or career set in their settings and enough energy, you can go ahead and start normal training them in the selected pathway.   Foals that can be added to registries can then also go ahead and be entered into Halter Shows for early levels.   (In this case you would want them registered into your APH related campaign if its already created and approved).    However if the foals are a crossbreed without a registry or a grade they cannot compete in halter shows and will have to wait until adults to be discipline showed or careered.   

If you have any more questions or need a part of this explained, feel free to let me know!