I don't understand acceleration and how something can accelerate without speeding up.

For this we must take a step back and analyse the meaning of acceleration. Acceleration is a change in VELOCITY - this is very important. Velocity is speed in a particular DIRECTION. You may be confusing velocity with speed which has no direction.

An object can accelerate without actually increasing in speed - it may just be changing direction. For example, a satellite in orbit around the earth is always changing direction as it moves in orbit, but it stays at the same speed. We say that it is constantly accelerating because its velocity is changing. Velocity can change if EITHER the direction changes OR the speed changes (or both).