If you're curious to learn more about this, I really strongly recommend reading the latest IPCC Assessment Report Summary for Policymakers[1]. It's a pretty easy read, will take an hour or two, and you'll come out the other side with a really good understanding of the fundamentals.
[1] Here's the latest, from AR5 in 2014. It's a little out of date, but it holds up fine as an introduction (mostly what's new since then is "we've done nothing, so things are now even worse"). AR6 is due out in 2022. https://www.ipcc.ch/site/assets/uploads/2018/02/AR5_SYR_FINA... (3.5 MB)
Rainfall will be different. There will be more or less rain, at different times of the year, with a changed susceptibility to droughts and/ or floods. So yes, there may be more rain overall but - just as some places may get cooler - local climatic changes will be in multiple directions on multiple axes.
It won't be a smooth change to the new conditions either; there will be centuries of instability and infrastructure and agriculture will have to adapt to obsolescence every few decades.
I'm not a climatologist but I do know some of the reasons why:
The thermal gradient between the equator and the poles (a major driver) will weaken - the poles will heat up more quickly.
Ice at the poles drives some ocean currents.
Temperatures at different heights in the atmosphere will be affected differently.
Increased evaporation from oceans but higher ambient temperatures will change cloud formation unevenly.
Nights will cool down more slowly due to the 'blanket' effect of CO2.
The overall effects are interconnected of course, which is why there is so much work on modelling and so much work still to do.
Edit: I forgot to mention the effects on vegetation and landscape that will also feedback unevenly into the climate. e.g. rainforests depend on and generate a lot of rain; a rainforest may hit a tipping point where it is no longer viable and rainfall will then collapse over a large area.
To add to the other poster - the type of rainfall will change as well. There is an appreciable difference between the spring showers we used to get that rained 2 or 3 inches over a 12-14 hour period, and the gully washers we have gotten in the last five to ten years locally that drop 1/2 to 1 inch in an hour or less.
Hotter air can hold more water. In cooler climates, the air over a region will become saturated with moisture, causing precipitation. In warmer climates, the air is able to hold a lot more water before becoming saturated, therefore the rain falls somewhere else.
I imagine you'll get more rainfall over oceans; air currents over land will hold onto moisture until they blow over oceans, cooling them. I also imagine you'll get more rainfall on the sides of mountains which are already regions where warm, damp air rapidly cools causing precipitation.
Please don't take this as truth, I'm just theorizing, I'm not qualified.
