Your carriage is what makes driving your horse fun. So you should make sure it’s in good running order. Fortunately, carriages aren’t terribly complicated machines. After all, your horse is the motor (which is a fairly complicated machine.)
Here’s an extensive checklist that will cover most of the parts of most of the carriages. Carriages vary quite a bit from one to the next, so there can’t be a single comprehensive list that would cover every carriage, but this list comes close. Go through each of your carriages with this list at least once a year.