While on the mat, execute your movements only as quickly as you can while remaining in a relaxed state. There is rarely a need to rush. Speed will come naturally with proficiency. If you train at a faster speed than is appropriate for you, then you are like a skier who is racing down a hill too steep for his ability: an accident waiting to happen. Training too fast also reinforces mistakes which would be corrected at a slower pace. Speed will come with time and careful repetition of techniques. Take it easy and be precise. In this way you will learn well.
When making fine wine, timing is critical. The finest grapes and other ingredients will be wasted without the winemaker's knowledge, skill, and good sense to leave the wine in the cask (later the bottles) until just the right moment. Likewise, each aikido technique has its own proper "time", and must be permitted to mature. Carefully study the timing of each technique. You will discover that each has its own rhythm that cannot be rushed. The interaction between you and your partner has a universal rhythm, like the tide, and you must know when to flow in... and when to flow out. The essence of your practice is to study this rhythm, for it is the rhythm of life.
Ki is the Japanese word for spirit, energy, life-force. When ki is usefully active in your body, it is known as kokyu-ryoku or "breath power". When you breathe freely, your energy can also flow freely so you can experience ceneteredness, relaxation, and extension. There is hardly a moment in which you will need to hold or restrict your breath in aikido. We do engage occasionally in special breathing exercises in which we hold our breath, but these have specific purposes related to your breathing capacity and the stretching of your mind/body. Always maintain a continuation of your breathing process when you practice your movements. All your movement techniques are easier when you breathe freely. This is especially true when rolling or taking falls: always exhale as you contact the mat during rolling and falling.