That depends on the region. As a well-known example, rainfall in California will decrease.

Here is an explanation. Take a pot of water. Heat one side. The water will start to circulate, come up where it is hot, fall where it is not, then cycle back.

The Earth's atmosphere does that as well. It rises in the tropics, spreads out, falls in temperate zones, then comes in. These circulation patterns are called the Hadley cells. They explain a lot of things. For example air that came from the equator, then spread out, is moving faster than the ground below it. That's the jet stream. And where it falls it is still moving. That sets up the prevailing wind on land, and the trade winds at sea.

One thing that they help explain is where things are dry. You see when air rises, the water falls out. So where air comes down, it is dry. Therefore the latitudes where it falls have lots of deserts.

Now one of the effects of global warming is that the Hadley cells get less stable, and also expand. Which means that arid zones in Mexico move north, and the Southern USA gets drier. But this comes with more instability in weather patterns.

So you see, it is complicated. Overall wetter does not mean wetter where you are...