You need to have a way of connecting things mechanically (e.g. screws) and electronically (e.g. wires). Depending on the connectors that you have, yes you may need to solder something.
If you want to build a robot like that, you would need to have a basic understanding of electronics. Then you need to take the datasheet of your components, look at their electrical characteristics and make sure the circuit you make meets their needs (by carefully adding resistors, capacitors and inductors). An example from the driver you have picked:
If you are lucky, the datasheet may have a "typical usage" you could gain insight from. Example from your driver:
There is also connection to the microcontroller, which again means you should take the datasheet of the micro, look at what pins are available for what, and then decide which of your component pins should connect to which of the micro's pins. For example, you may need an interrupt from a certain component and therefore you should make sure to connect it to a pin that can generate an interrupt.
You should also pay attention not to break the micro. For example, you shouldn't try to power a motor directly by a pin of the micro because usually the current that the micro can give out at its pins is not enough to drive a motor (and you'd risk burning the micro). You have done well by buying the motor drivers.
Finally, you need to program the micro, which depending on the microcontroller may be widely different. Arduino has tried to make the coding as simple as possible, so if you don't try to do smart things and stick with the basics, its libraries should be fine.
Tip for things to go smoothly: read the datasheets carefully.