A lot depends on what "era" you are looking at. A Formula 1 race car from the 1930's could be built relatively easily with fairly simple technology, metalwork for the body, mechanical brakes, leaf springs, etc. I am assuming that the power is supplied by some sort of magic, since building an internal combustion engine will require far more skillsets and "tools to make the tools" than likely possible with only a month.
1930 era F1 car
Post war cars into the 1960's have much more advanced systems, including independent suspensions, hydraulic brakes and are starting to do serious investigation into aerodynamics, and often have elaborate lightweight "birdcage" frames made up of welded tubing. Once again, this requires more skillsets (adjusting the suspension is a new skill which was not as evident in a 1930 era car, for example). Engine technology was also increasing rapidly, including fuel injection.
The jump from the late 1930 era Auto Union to...
...a 1960 era Ferrari is just tremendous
The 1970's and 80's really saw an explosion in technology, from aerodynamics to active suspensions to turbocharged engines, as well as significant improvements in driver safety. Chassis construction moved to monocoque "tubs" and incorporating the engine block as a stressed member to make the chassis stronger and lighter. We actually don't see many of these improvements today, as the governing bodies banned or severely restricted many of these technologies in the interest of keeping the races competitive and smaller teams (without the sorts of research budgets that Lotus, Renault or TAG could bring to bear) viable.
1987 Lotus 99T with active suspension and aerodynamic "tunnels" under the chassis
Modern cars are essentially computer operated in many aspects, and have devices like KERS (Kinetic Energy Recuperation System) to recover some of the braking energy and feed it back when accelerating out of a corner. Team mechanics can monitor hundreds of channels of information about the car as it is running, and although the current rules do not allow the car to be accessed and modified electronically in motion, it is theoretically quite possible to adjust the suspension or angle of aerodynamic aids based on these inputs as the car is in motion.
So your team will need to know and understand a great many different technologies, materials science and aerodynamics, as well as engine technology, in order to build something as complex as a race car, and more importantly, to tune and adjust it for peak performance on any particular track.