Most of the driver expenses on the road will be covered by the trucking company. Fuel, tolls, truck maintenance and the like will be the responsibility of your company.
The drivers main expense on the road is going to be food. Most drivers will plan to bring food with them in the truck, as eating out every day gets pretty expensive, pretty quickly.
Additionally, any tickets you receive that are charged to the company, i.e. overweight tickets, you will generally be required to reimburse them for. Out on the road, driving a legal vehicle is solely the responsibility of the driver themselves. If it's not legal, you don't drive it.
Article - Making it Financially in Truck Driving
Finances on the road can be tough when first getting started, but hopefully this will provide you with a little insight on how to stay out of trouble.
Forum - Expenses on the road, what expenses?
You will be responsible for feeding yourself - that is the biggest expense while living a traveling lifestyle. You can learn to reduce the costs by keeping a refrigerator in your truck and learning to make some of your own meals in your truck.