If you are just needing a new one then everything else should be able to stay the same.
Do you definitely need new chainrings? They last pretty much forever unless they're damaged, in my experience. But nonetheless, really all you need is the BCD which is the measurement of the bolt distances:
https://www.wolftoothcomponents.com/pages/how-to-measure-bolt-circle-diameter-bcd
Then you need to know how many teeth your current rings have. Then order the same spec and install.
However if you are going from a single chainring to a triple then yes you will need a wider bracket as well to account from the extra rings. Assuming a square taper bb, it will only have two measurements, the shell, which will be either 68 or 73mm and then the total width (e.g. could be 110mm or 122mm or anything really). The latter will determine the clearance you get for the rings from the chainstays. It is measurable and your LBS could do it. This is ultimately going to determine your chainline, which you want to be as straight as you can (I.e. middle ring up front in line with middle ring on the cassette).
I'm by no means an expert but this is what I'd be looking for