The major factors in temperature change from precipitation...
, fundamentally rain is falling to the ground from higher in the sky.
Precipitation typically comes from a location where it is colder compared to us on the ground (the troposphere generally is cooler with height, on average around 6.5 Celsius\km.).
Basically, the rain itself usually at least feels colder than the air beforehand, right??
Well, if you do bring something colder to a warmer environment, their temperatures will tend to move towards a balance between the two.
[++ see Second Law of Thermodynamics ++]
So with significant rainfalls, there's a lot of cold water being spread down into our warmer air, and due the dispersed nature of rain (a bunch of small drops rather than one giant stream) it ensures that rain is quite quick and effective in spreading it's cold temperatures into the air.
, falling rain also encourages downward air movement, due to friction.
[++ see downdrafts
So not only is rain (that is almost always colder) coming down, but air (that is almost always colder) is coming down too.
Furthermore, rain is cooling the air it is falling through, colder air sinks, so the downdraft is further enhanced.
[++ Actually the air must be quite a bit cooler to sink on its own... see adiabatic warming ++]
So this is a second fairly quick cooling process, as downdrafts occur within the lifecycle of a thunderstorm cell/complex.
And third, some of the rain/snow/sleet/hail/etc water will begin to evaporate (or sublimate!) as it is falling and once it reaches the ground, unless the air is totally saturated. When water evaporates, it needs to gain energy. To do so, it takes energy from the environment, namely the air.
This evaporative cooling just exacerbates the first two influences above. And furthermore can continue to have impacts days after the rainfall compared to drier places, stunting high temperature ascension during the afternoon.
[There are also other factors important in the longer term, such as snow reflecting sunlight (cooling the area), moister air being favored for cloud formation (which cools during the day, warms at night), and the fact moister air changes temperature at a slower rate due to its heat capacity (cooling during the day, warming at night).]
But, the result is, scientifically, when precipitation falls, it is fundamentally a cooling process.
The stronger the precipitation is, the much more significant the cooling. And certain atmospheric setups enhance the cooling effect even more (steep lapse rates, dry air nearer the ground)
Now, even after saying all that, it actually can still be warmer soon after rainfall.
How in the world is that possible!?!
Because wind can still blow in large amounts of air from other areas... sometimes from environments that haven't yet themselves experienced significant precipitation.
This often happens as warm fronts pass.
[++ Heaviest rainfalls do more often come with cold fronts, which bring in more cold air. However most (nontropical) lows start with a balance of areas of advancing warm air followed by surging cold air. So you may well see both... or you may not: in more mature storm systems - as you'll often see in winter - most warm air is completely undercut by cold air... and so many large areas never experience any notable warmup as a storm system works through ++]
Other unbalanced factors can overwhelm the cooling effect as well... like when it's cloud all day, then rains a bit, but finally the sun comes out. The hot sun can quickly turn the battle back towards the warmer side.
[++ There is indeed at least one other very very rare way, where if the atmosphere is setup perfectly, a strong downdraft can actually begin to warm due to adiabatic compression, but still have enough oomph to reach the ground... and you wind up with a heat burst. ++]
So air after rainfall is cooled in the extreme majority of cases... it's just that your cooled air may instead end up being relocated to other locations.
Check out the topic of outflow boundaries
So, to your questions:
⇒ Will it be colder right after a storm compared to how it was
right before the storm?
⇒ Will it be colder the day after a storm compared to how it was right before the storm?
Depends what the weather pattern and air mass movement is like. If a cold front moves in, yes. If a warm front moves in, no. In the tropics or during the height of summer, it will often be about the same the next day as the atmosphere mixes the cool air from yesterday's rain around until it's spread out and minimized, and the air heats back up again the next day when the sun comes out.
⇒ Will it be colder after a storm than it otherwise would have been without the storm?
Basically yes. A nearby site that doesn't get the storm's effects will almost certainly be warmer.