Chariots, LOL, I want a chariot. Well dude, sorry to disappoint you, but they WALKED as amazing as that may seem. Unless you were a child or old guy with bad legs, in which case you rode an ass. The problem with asses is that they are as slow as molasses and they can get ornery if you overburden them. Like horses they are insanely uncomfortable if you ride them more than an hour or two.
Before saddles were invented, horses could only be ridden relatively short distances, and in any case, were only for the rich. Try riding a horse for a few hours with only a blanket or bare back. I guarantee you will not repeat that experience. In fact, if you can ride a horse for more than an hour with only a blanket or Roman saddle you are a better man than I. Modern saddles, which were originally developed by the Mongolians, have a very special frame and padding of ingenious design, and stirrups, The Roman saddles were greatly inferior to modern saddles and much more uncomfortable to both the rider and the horse.
Even with a modern saddle, long distance travel is uncomfortable. My great, great, great grandfather, who was in the 7th Cavalry rode all the way from Texas to Pennsylvania on horseback with no rest after his tour of duty was up, and this was considered a feat of endurance worthy of his badass reputation.
REALLY rich people got carried around on litters, called lecticae.
You could also travel by cart, drawn either by oxen or horses, but this was a VERY jarring ride because the carts had no springs; basically an act of desperation. Generally, people only did this if they were so sick they could not walk. Riding in an axel-only cart on a cobblestone or dirt road will make you wish you were dead.
Chariots are actually a pretty good way to travel, but you have to stand up, and they were very expensive. For this reason they were more of a war vehicle for the elite rather than a travel device. Romans did not use them significantly.