The trouble is, just talking about something can be perceived as bragging. If you buy a new car, or something for the house, or go on holiday, or if, like me, you are proud of your grandchildren and talk about them, I don't see this as bragging exactly. I call it talking about things that are of interest to you.
Most subjects can come under the umbrella of bragging. If you suffer from an illness or injury, it's natural enough to talk about it.
What subjects can people talk about which are not perceived as bragging?
Most people, whether they admit it or not, are mainly interested in themselves and their loved ones. It's polite to listen to other people talk about their own lives, but how many of us can truly say we are really interested in what others tell us? We may be momentarily interested or concerned, and have a reciprocal conversation, but deep down, we are concerned about our own affairs.