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Using solid fuel in the first stage of rocket flight is characteristic of parallel rocket staging. The parallel setup uses solid rocket fuel during takeoff because solid fuel is very stable and provides a continuous thrust until it burns up. Liquid fuel provides variable thrust that can be controlled, which is better for later stages of flight. 

The main reason for the multi-stage rocket design is to keep the rocket as light as possible during flight. After a rocket has used up a certain amount of fuel, the structure and engine that used that fuel becomes useless to the rocket, and slows it down. Stages allow a rocket to drop these used-up sections. A good example of a multi-stage rocket is the Saturn V rocket.