If you have the plan set and all the permits you can dig the foundation and poor the foundation. Next you frame the walls. After framing and erecting the walls one sheets the roof. make sure you get all the required inspections
first you need permits. then a foundation for it (concrete) and then framing and electrical and finally interior and exterior wall protection (drywall, paneling, vinyl siding)
Put up the exterior walls first before demolishing any existing walls. This will ensure that the weather conditions and animals are kept out of the construction space.
It isn't as easy as it seems. You need to get an engineer to approve of your addition, then you have to take his approval and fill out the paper work at the building commissioner. Then you have to hire an inspector at various points of the build.
First you have to decide of which material you want to build your addition. Wood, brick, concrete? Oyou also have to remove existing walls en possibly roofs to connect.