The phrase "go around ___ing" describes someone doing something bad again and again, with different people.
For example, if someone has been spreading false rumors about you, you can say:
She's been going around telling everyone that I slept with Kyle. That's a lie!
If your husband or wife has been spending a lot of money on useless things, you can say:
You think you can just go around spending money left and right? We've got bills to pay, damn it!
It is possible to use "go around ___ing" to talk about someone doing a good thing many times, but it's rare. Even if you're talking about the person doing something good, it might seem like you're complaining:
He's always going around telling everyone what a great job they're doing.