If my carriage can cover about 28 miles in an 8 hour day of travel because the roads are in very good conditions, how fast could someone on horseback take? How about someone walking? Would the speed of this person be different if they’re not used to riding or walking on a regular basis?
It's really not possible to give a definitive answer, since it depends on the fitness of the horse, or the person walking/running.
At an extreme, both elite & well-trained horses and humans can do 100 mile endurance races on mountain trails in 12-17 hours (Tevis Cup winning times: http://www.teviscup.org/results/tevis-cup-winners Western States 100 times: https://en.wikipedia.org/wiki/Western_States_Endurance_Run#Results Both races are run over the same trail.)
So from simple math, a very well-trained runner or horse & rider could do that distance in 9-12 hours. Someone who is basically fit but unused to walking any distance might take a week or so. An untrained rider on an out-of-condition horse might take even longer, as the horse will need rest and recuperation too. (Even doing much shorter rides - 20-25 miles at most - I give my horse at least a day off between rides.) An untrained rider is also going to be in considerable pain after the first couple of hours.
Of course, this is for an individual. If switching horses frequently, a person could cover distance much faster. E.g. the American Pony Express did around 10-12.5 mph, changing horses every 10 miles: https://en.wikipedia.org/wiki/Pony_Express
Fastest way to move is to ride horses like hell and exchange them every few kilometres on a horse change station. Assume you can make 30km per hour like that.
It then just depends on your personal fitness how far you can get.
Oh and yes. You have to be rich enough to set up those horse change stations on the route. Wow that's huge cost! Imagine in modern world words, buying a car for every 15 km you want to travel and let them park and get maintained by someone.
I think it's a thing you could do to send messengers to your army from the capital and back while you're at war, but for anything else it might be too expensive.
Counts Thurn and Taxis did set up a subcontracted carriage relais system around 1490 or 1500 on most of the important routes through europe for fast messaging and travel. It was, in the end, the invention of the postal service.
People on a good road made between 10 and 15 kilometres a day. That's not because they were lazy. You could think "eh after all, I can make 15 kilometres in 3 hours easily". It is because you have to transport all your stuff on your back, and you have to sustain the speed for 10 days. (38km x 3 divided by 15, roughly 10)
If you exhaust yourself the first day, you won't make any advance on the next two days, losing more than you win. Add a suboptimal spacing of sleeping taverns and you rather decide to stay here early in the afternoon than to risk the next house. No streetlights also means there is real darkness, you don't want to risk that.
That said, after having read Duncan Drake's comment, I checked Wikipedia for the roman legions. They indeed made 30km in 6 hours according to Wikipedia, and that was their day's march. Of course, they were trained and well-fed. I can't imagine that everybody can do that.
I am currently building my fitness up back from a chemotherapy-induced nothingness. I do it by walking. Normally I am a lazy bureau-sitter. 😊
During the chemotherapy, on the lowest level of fitness I ever had in my life, there was 2 km maximum per day. It just wasn't possible to do more.
Bureau without sport, 1 year later: 8-9 km at a speed of 4km per hour. I was so darn exhausted in the evening that I would fall asleep sitting.
Today, not really sporty but building up towards something healthy: I do 12-14 km each day, speed around 5 -7 km per hour. Not specially exhausted in the evening. (But I am not wearing my household on my back)