I know that it's been a while since I've posted because I've finally gotten around to my story and there have been several changes to it as well, such as the title, which from now on it's going to be called The Celestial Knights: Tales of Agravain and I have finished the prologue about a week ago, and now, I'm currently working on the first chapter and so far, I'm actually proud of what I've written, but right now, I'm stuck on what else to do for the intro of it and here's what I have so far: The clip-clop of horse hooves trotted against the cobblestone streets as wheels thumped on the pavement... That's what I've got so far, but I'm stumped on what else to write. My MC, Kamaria Imani Rosita Igwe De La Cruz is going to school and I'm not sure how to flesh it out even more. The plot's still in the works, so I'll show you guys what I have when I finish the book.
I'd start with the sounds of the carriage and the horses like you did. (sensory) Then the sounds of the driver. (sensory) Maybe even the smell of the driver and how disheveled he was when the MC & co boarded the carriage. Use that to springboard to the MC's thoughts about the driver and his class. (pulling into memory/imagination, it should not be immediate but looking at the bigger picture) That gives you a chance to build character. Then I'd refocus the MC's attention out the the window at the passing scenery for historical details. Most of the above is setting, so logic says that you come out of that with dialog referencing the plot or something close to that.
My two cents worth: I get sea sick riding in enclosed carriages that are suspended from leather straps. I suppose the straps act like shock absorbers, but the swaying destroys me. Maybe Kamaria has inner ear problems?
Round wheels don't thump on pavement. I'd suggest something akin to "rumbled" for wheels rolling over cobblestones. As Catriona commented, old-time coaches didn't have springs. The coach was suspended on the axles or chassis by leather straps. Leather straps often squeak as they move. There would definitely be a swaying motion to the ride. You could mention a horse snorting. Sights? Village sounds and smells? Wood smoke from morning fires as the people prepare their breakfasts?
It's going to be loud. Assuming the horses are shod (have shoes on), it's going to be even louder. Personally, I can't think when working on pavement or packed gravel with a fully shod or half-shod horse. The class clip-clop sound is produced only by new shoes, and it doesn't take long (-1 week) for them to become scratch and produce a more grating sound. Anyway, outside of my pet peeve about horses, I agree with a lot of what's already been said. Ridding in a carriage is going to be uncomfortable and bouncing, so I'd say your MC is probably holding on to something and very ready to get where they are going. Smell will be your best, it sets the era and location of a story better than anything. Has one of your MC's fellow passenger been sick? Is someone wearing heavy perfume? Those are just to examples out of many, I hope you the best of your work.