They are both correct and in conversation many people will find them to mean the same thing. However, technically there can be a difference. Here is an example. He started traveling through Europe last week. This is the past progressive. When I hear or read this, I assume that he is still traveling through Europe. With your other example - He started to travel through Europe last week, it sounds like he started to travel but had to stop. In casual conversation, people will understand what you mean if you use them interchangeably. However there is a difference.