As for boarding, somewhere on your horses page there should be text saying "(horses name) is not registered for in an equestrian Centre" theres a little arrow/link to the right of it usually that will take you to a new page, showing you different centers.
But don't just grab the one with the highest prestige. There should be a filter option on the page next to the search bar. Adjust the settings and requirements you need. I'd recommend starting with a centre that charges 20e/day and offers you full tack to start with, and as you learn the game better, start asking for different things.
The best time to find a good EC is right after the game updates, because all the old tenancies end and anyone that didn't extend their stay has their horse kicked out. Once you have a center you like, you can extend for a month at a time, or you can just place horses in which ever stable is available when you want to work the horse (thats my method because I don't work the same horses every day)