To buy a car you need to enter 'build/buy' mode then choose 'buy'. Now scroll down to 'vehicles' and under this tab you with be prompted with either 'bikes' or 'automobiles'. Since for this achievement we need a car, choose 'automobiles' choose your car and buy it like you would any other item in the game.
HOWEVER!
Before you can even place your car into your sims' world you need to purchase a parking spot which is also located in the 'automobiles' section. It costs 150 of the sim currency.
The parking spot must go in the lot surrounding the house, it can't be placed outside of the lot such as on the road etc and it can't intersect other objects or be placed on un-even ground.
The parking spot can be decorated using the terrain tool, so you can cover it with grass, stone, wood etc and still park your car on it.