How to use the system to travel? That's indeed a really interesting question. Because the traveling system is so highly develop it would be a shame to don't use it to our own advantage.
One of the easiest and most famous one is simply to do hitchhiking.
You stand on the side of the road or at a petrol station, you hold a carton board displaying your destination, you raise your thumb. Perfect, now you only have to smile and wait for a driver to stop and pick you up!
This system rely only on strangers helping each other.
Hitchiking is mainly for free but some cultures expect a money exchange for the petrol.
Make sure to check the country laws about hitchhiking first as it can be illegal;
Please, be very careful. It's a safe way to travel but you should really be careful about who is picking you up. Trust your instinct, don't go in a car if you don't feel good.
Car sharing is quite similar but the difference plays in the fact that it's arranged in advance. Either because you know the person or you've been in contact before. So you define a place and time to meet and then share the road together. It's also quite common that you'll pay a fee to cover the travel expenses (toll, petrol, ..).
You can find car sharing websites online easily. But it doesn't exist everywhere, in which case, hitchiking.
Inside the cities themselves you should also prefer the common transport solutions like buses, subways or trains. And if you can, go for a bike or simply walk.