I am assuming that the upper wire is the cable that goes to the ceiling where the fan/light is to be installed. If so, you should have a red wire in the ceiling box that is unused and probably capped with a wire nut.
Your description of the switch box seems a little off. If all three black wires were connected together, what wires go to the switch?
More than likely, one of the lower black wires is the hot source, and it is connected to the other lower black to power another box, and is also connected to the existing switch. The upper black was probably also connected to the switch asn was hot when the switch is on and not hot (open) when the switch is off. The whites are all neutrals and all should be connected to each other (and not to the switch).
You can install a double switch or two separate switches in the box to separately control the fan and light. The lower hot black goes to the common on a double switch or one pole on each of the two switches. The upper black goes to the other side of the first switch and the upper red goes to the other side of the second switch. Effectively, these switches share an "in" but have separate "outs".
In the switch box, all of the whites (neutrals) continue to be connected.
At the ceiling box, the white goes to white, the black goes to the wire for either fan or light, and the red goes to the other. The fan wire colors may vary, but the instructions should indicate which is which.
You make no mention of green or bare wire (ground). In a modern, properly wired system, there also should be these, both from each cable and at the fan. Ground wires are connected together and to the base of devices, switches and metal boxes or fixtures. If they are present, connect them. If they are not, you have an ungrounded system that poses a bit of risk if a device is damaged or shorts out.