Two Ways to Accelerate
Let me give you two examples of different techniques for accelerating on your bicycle. These techniques aren't just for racing or group rides; you can also use these methods of getting up to speed when fighting a headwind or when rolling away from a red light or stop sign. While these are only two examples and many more exist, these two are a good starting point experimenting with what you prefer and what you are weakest at and need to practice more.
1. Long, big, and steady; downshift at the end
Start in a bigger gear than you settle on at the end of the acceleration
Begin slowly out of the saddle and use more pedal strokes to accelerate steadily
Find a place to "hold" your speed, sit down and downshift if more cadence would help you sustain your speed
2. Quick, short, spike; upshift at the end
Begin in a smaller gear and either hold cadence or shift up at the end
Start with a quick hard acceleration, sitting or standing
Max out the gear by spinning out your legs. The acceleration is over when you reach max cadence. Shift up until you find a cadence you can sustain.
You can pile several of these quick accelerations on top of one another to get to a higher max speed.