Just note: These are modified rules of RaW that you can find on page 181 of Players Handbook 5e. I like to over complicate things, so...
When characters travel longer distances you calculate distance traveled based on their movement speed. Group can only travel as fast as its slowest member (this also includes instances when some members have their speed reduced, i. e. by spell, difficult terrain or exhaustion).
Characters need to choose a Travel Pace. While traveling at a slow pace (1/3 slower), characters have the ability to move stealthily and avoid detection. At a fast pace (1/3 faster) characters have -5 penalty to passive Wisdom (Perception) scores, but if you want your players to roll for things, disadvantage to Wisdom (Perception) checks is also applicable. For character with speed 30 Travel Pace looks like this:
| Pace | Speed | Per hour | Per day |
|---|---|---|---|
| Slow | 20 ft. | 2.4 miles | 19.2 miles |
| Normal | 30 ft. | 3.4 miles | 27.2 miles |
| Fast | 40 ft. | 4.4 miles | 35.2 miles |
So a character with speed 30 ft. using normal pace can travel 27.2 miles per day. This assumes that the character traveled only 8 hours, because any further travel becomes Forced March. Characters can choose (or be forced to) travel longer than 8 hours per day, but after each hour they need to make Constitution check (or a saving throw in RaW) DC 10 + each hour traveled past 8 hours. On failed save character suffers one level of Exhaustion.
Mount, with some exceptions, can only move at a normal pace. Trained animals like post horses, messenger griffons and others can move at twice the normal pace for one hour, but any further travel becomes Forced March. Forced March rules apply to both mount (only to creatures that can become exhausted) and rider, but also require rider to succeed Wisdom (Animal Handling) check DC 15 + each hour traveled past 8 hours. Magical and mechanical mounts like brooms and carpets always travel at normal pace and can travel a whole day without stopping.
| Mount | Speed | Capable of fast travel |
|---|---|---|
| Mule | 20 ft. | - |
| Donkey | 20 ft. | - |
| Pony | 30 ft. | Yes, after training |
| Draft horse | 40 ft. | - |
| Riding horse | 60 ft. | Yes, after training |
| War horse | 40 ft. | Yes, after training |
| Unicorn | 50 ft. | Yes, if reasoned with |
| Pegasus | 60 ft., fly 80 ft. | Yes, if reasoned with |
| Wyvern | fly 80 ft. | - |
| Griffon | fly 80 ft. | Yes, after training |
| Roc | fly 120 ft. | - |
| Dragon | 40 ft., fly 80 ft. | Yes, if reasoned with |
| Flying broom | fly 50 ft. | Never |
| Flying carpet | fly 30-80 ft. | Never |
| Robotic ostrich | 60 ft. | Never |
Vehicles can move only at normal pace and their speed is the lower of either their own or speed of the animal towing them. A mule with speed 20 ft. towing chariot with speed 60 ft. have speed of 20 ft. Some example vehicles with their speed can be found here:
| Vehicle | Speed |
|---|---|
| Cart | 30 ft. |
| Coach | 40 ft. |
| Chariot | 60 ft. |
| Flying city/fortress | 10 ft. |
| Air ship | 80 ft. |
| Flying war chariot | 120 ft. |
| War ship | 35 ft. |
| Transport ship | 60 ft. |
| Pursue/Pirate ship at full sail | 80 ft. |
Driving a vehicle requires proficiency and Forced March rules apply only to creatures operating the vehicle and mounts towing it.