Roads not only speed up movement, but they make connected cities more efficient (I think but I’m not sure it is just the money.) For this latter purpose, they are only needed if the city is not already connected back to the capital by river/water. To build, you must be adjacent to a city or to a road tile that connects back to the city.
I think it depends very much on where you are in the game. Early on, the problem is usually wood, because that takes longer to research. (Although you can have your workers chop, in or out of your territory, to tide you over.) But if you expand quickly, food can be an issue very soon. If you expand infantry quickly, iron becomes an issue. And to play above the lowest difficulty levels, you need to be aggressive with buildings, such that it is not too long before you always have a desperate need for stone. (Also for wonders!)