Carb icing IIRC was an important hazard to watch for even if it wasn’t raining - a function of temperature and humidity. Students were warned to recognize the signs of diminishing power and apply heat. Carb heat on was a standard thing on approach - if you suddenly decide not to land, you likely want full power and a constricted carb is not good. OTOH, we were also warned of the famous rookie mistake - you put on carb heat, the ice starts to melt, water through the engine causes it to run rough, pilot thinks there’s a problem and pulls off the carb heat instead of letting it finish cleaning out the carb throat.
If it’s wet out and cool, carb heat is a good idea.
You stay away from thunderstorms mainly because of the wind currents. The same currents that can keep juggling golf-ball sized hail chunks (or larger) could rip the wings off a small plane. One of my instructors related a story of flying a Grumman Tiger through the edge of a storm and said the ends of the wings were flexing well over a foot - he was glad to get on the ground.
I never got formal hood training for a Private license; but my instructor did take me into a cloud and ask me to try to fly straight and level by gut feel. After a minute he said, “look at the instruments now”. We were in a descending spiral, what felt “level” was actually a banking descending turn. If the cloud had been really low, we might have seen the ground too late to recover. Instruments are very important.
However, the pitot, a critical air speed indicator, can also ice up. In smaller planes, some instruments are powered by the pressure differential and if that hole freezes up, you lose critical instruments. No instruments is not good if you encounter low visibility. (A frozen pitot in a thunderstorm was apparently what triggered the Air France disaster over the south Atlantic. The autopilot had no indication of air speed.)
Remember, too, that somewhere between your summer rain and the cloud full of ice hail is a point where the water will freeze on contact, so pilots have to pay special attention to wing icing. Ice changes the shape (and smoothness) of the wing and hence its lift, and several square yards of ice is a bit too much extra weight. As for flying through hail - I’ve never even seen a warning about that, if you reach that point you’ve got plenty of other problems.
But flying through rain? If I didn’t do that, I’d never have gotten my pilot’s license. Several times, I remember flying where you could see for 50 miles or more the showers marching across the landscape like the occasional transparent curtains, and nice sunshine in between.