Transportation during the Regency-[late 18th century to early 19th century]

There were no 'planes, trains or automobiles' in Regency days so the vast majority of the populace walked to their destinations.
If they could afford a horse, they rode:

The heavy horse (above right) was the key to the myriad of carts, wagons, coaches and carriages in the Regency world.
Transportation was based on whatever that heavy horse, in team with others, could pull.
The two coaches above are both Austrian.
The Landau on the left dates from 1825; the Phaeton on the right is earlier dated to 1815
There are few Regency coaches extant. There were major changes in coach design in the mid-1800's
and older coaches were rapidly discarded or altered to meet the new, more comfortable designs.
Below is a useful illustration of coach basics, after the developmental of the elliptical spring.

A curricle
A phaeton (detail from a Stubbs painting)
Two-wheeled carriages in woodcuts
A bretcha
A landaulet
A light phaeton
A travelling chariot
A coach and six
A post chaise
A variety of carts, gigs, tilburies and whiskies were becoming available.
Here on the left, a cocking cart, and on the right a curate-cart