The Travel Agent is where you can travel to other cities within the country you are in. You are able to travel by bus, or if you own a car, you can travel at a discounted rate by car.
You will not be able to access all cities in the game when you first start. Most have level restrictions for entry, some require you to be a Respected Mobster, and others require you to have a certain number of Gold Medals before entering.
You will be unable to travel if you are in Hospital or Prison, so it is wise to stock up your inventory with appropriate supplies before travelling to areas that don't have the shops you require.
The Airport is where you can travel between major cities around the world. Each country has one main city with an Airport that is always open, other cities are required to have a certain population before an Airport is opened there.
Just like travelling by Bus or Car, you are unable to use the Airport if you are in Hospital or Prison.
In the Airport there are lockers and it's well known that one of the lockers contains a large sum of money. Each day, you have 10 chances to try and guess the combination to a locker for the chance to grab the money and run.