An object moving in a circle, is by definition, undergoing acceleration. Acceleration is a change in speed or direction. You do not have to slow down in order to move along a circular path, although you might want to if the tires on your car are bald or there is water on the road. :)
The example of an object going up and down and having a constant acceleration is exactly correct. Gravity acts at 9.81 m/s/s on the object regardless of the direction it is moving. On the way up, the object is slowing (assuming the only force acting on it is gravity) and on the way down it is speeding up.
This is a good question and requires some thought. I'm going to steal it for my classroom!