Question

Why does water heat up and cool down slowly?

Answer 1

Because water has such a high "specific heat" ... it takes so much heat energy to raise the temperature of a pot or a glass of water by only 1 degree. There's only one other substance on Earth that stores more heat than water does for such a small change in temperature. (Liquid ammonia ... 12% more than water.)
That's one reason why, for example, we use water ice to cool our drinks ... it takes SO MUCH heat out of the drink just to melt the ice cube and then to warm up the melted freezing water that comes from it.
That's also why cities near the ocean or a big lake have cooler summers and warmer winters. In the Summer, the water nearby takes lot of heat away from the city and stores it, without getting full of heat. In the Winter, the warm water nearby pours out heat to keep the city warm, without running out of the heat that it holds.

Answer 2

Specific Heat Capacity. It has a high one so it takes a lot of energy to raise or lower the heat by 1 degree.