Should I switch to a 13-23T or 12-25T cassette? I am currently riding a Fuji Sportif 1.5 with original configuration. I am not using clipless pedals currently.

The bike come with 50/34T chainrings and a 9-speed 11-32T cassette. I noticed that most of the time I am using only 16-24T. Sometimes with 28T for climbing.

As a result, I am considering to switch to a 13-23T or 12-25T cassette. However, I am a newbie and do not have experience of switching cassette.

Are my thoughts correct? Moreover, are there other issues that I have missed? For example, compatibility with my bike.
Should work. Put a new chain on.   

New chain as a worn chain will wear the cassette as it is stretched and does not match up.
If the chain is severely worn it may even jump.   

Chain is cheaper and should be replaced unless it is almost new.
A cassette will typically last 2-3 chains.  

If you sometimes use 28 then why are you giving that up?
Is a closer gearing really worth giving up the range?
Why not like a 12-27T?