Why does Eustace want to ride his horse across the country? What is one significant thing that happens to him on the trip?

Eustace wants to ride his horse across the country because he wants to see how far he can push himself and the horses.
After that Eustace was stopped the other day by a family while he was riding his horse was when he realized that he needed an adventure and that leaving out in the country for a while would do him so good.

One significant thing that happens to Eustace while he is on the trip is that he buys Hobo, his prize horse. This horse was very energetic and fast paced just like Eustace and this is one of the main reasons that he decided to buy it. He bought Hobo while he was on the road across the country. Eustace is extremely happy that he now owns Hobo and there is a new side to him that we are starting to see.