Travelling rules

A 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 travelled 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).

Traveling by foot.

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.	4 miles	19.2 miles
Normal	30 ft.	4 miles	27.2 miles
Fast	40 ft.	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 travelled 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 travelled past 8 hours. On failed save character suffers one level of Exhaustion.

Mounts.

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 travelled 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
False Wurm	fly 80 ft.
Gryphon	fly 80 ft.	Yes, after training
Roc	fly 120 ft.
Drac Wurm	40 ft., fly 80 ft.	Yes, if reasoned with
Flying broom	fly 50 ft.	Never
Flying carpet	fly 30-80 ft.	Never
Maw Wurm	50 ft., Burrow 50 ft.	Only a fool would attempt

Vehicles.

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.
A hypothetical flying city/fortress	10 ft.
Air ship	80 ft.
Hypothetical 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.