Why does it get colder after it snows, and gets hotter after it rains?

Answer 1

I think you're thinking of the following:

Usually a snowstorm occurs on the northwest side of a storm (low pressure system), or along a cold front moving through your area from the north and west (assuming you are in the northern hemisphere). After the storm or cold front passes, the winds around the storm bring cold air into your area. Additionally, the snow that did fall reflects a great deal of the Sun's energy instead of absorbing it and heating up the Earth's surface like bare ground does.

As for the rain, you are probably thinking of after a thunderstorm or rain shower in the middle of the afternoon on a warm or hot spring/summer/fall day. After the storm or shower passes you, the sun appears to you again because the size of the cloud producing the shower/thunderstorm is relatively small. The sun quickly heats up the Earth's surface again, and many times it can seem like it never rained (especially if you were inside when it did and never saw it).

Answer 2

Previous answerer did well, only I'd like to add a few things.
In the south, the same winter low pressure front that brings snow up north will bring rain in the south. Winter rains in the south are a moderate rain that lasts all day, like the snow falling a few hundred miles north, only in liquid form. Afterward, the sky turns clear and the temperature drops, just like after a snowstorm in the north.
Summer rains are different, as the previous answerer explained.