It's done differently in different parts of the world. In Laos where I just came back from, trees are typically cut down with simple metal hand tools, sometimes even only a machete. They can be transported by hand or on a person's back but typically are transported in some sort of vehicle, like a wagon or a truck.
In the industrialized countries they're typically cut down using chain saws. Sometimes, especially when cutting them down near houses, they're cut down in pieces starting from the top. They can be transported in various ways. Typically they're moved by truck but they can also be transported for limited distance by very large helicopter or sometimes transported by boat or just sent down rivers.