Hi Blue!
Most of the time, the messages will say something like "+ X Energy" or "+ X Health" if you've just eaten, or "21 Wood" "5 Sap" if you've just chopped down a tree. Which often isn't necessary information, because you can see that you've just eaten, or chopped down a tree!
But occasionally, it will be something that you should take notice of. Such as (from Summer onwards) "A train is passing through Stardew Valley..." which can be useful information, although you'll also hear a loud "CHOO!", so you already know what's up just by listening.
There is also a message to say that the festival has begun, and it'll tell you where to go. But if you're good at remembering your mail, that's not vital.
The rest of the messages that I can think of all happen when you first wake up. These are messages that you should pay attention to. Some examples: "Your Apple Sapling was not able to grow last night", indicating that you should remove all the weeds/paths/machines near you sapling. "Your Copper Pickaxe is ready", nudging you in the direction of Clint.
If a rare event plays during the night, you'll get a notice for that, too. These are all coupled with a sound effect, though, so you probably won't miss it.
Very, very late in the game there are a few special messages that appear in a sequence, and while they're not necessary to gameplay, they are quite fun.
Hope this helps!
