Can you be more specific as to what you are struggling with?

I try to listen to the flavor of the chord, (aka V has a pull towards the root and I has a resolved feel) Others listen to the root, but personally I find this harder as some of the chords have inversions. I recommend familiarizing yourself with the feel of the basic I V IV chords by playing progressions on your instrument, and listening to how they sound. This website explains along with focused exercises: https://chordcrush.hooktheory.com/

Keep the root in your head and look out for where it appears in the progression, often you can work out the correct answer by process of elimation

For me is a mixture of things. There is the feel of the chords, then there are the basses of the chords (which I never try to really listen). Then there are the function of the chords, which is a bit more technical but helps to know how those things works.

In the end if I would give a tip to get better at chord progressions, listen to music and try to learn by ear. Maybe best start with children´s songs. When you are in a level where you can almost catch the chords as the music goes (upon knowing the key), then things will make more sense.

I think the best way is to take your favorite songs that you are super familiar with already and look up their progressions. Better yet play along with them, if you play an instrument. To Daniel's point, see how you feel after a particular chord change. After doing this for some time with songs you already know and love you start to hear similar patterns in all songs and genres.. i.e. The IV-V-vi cadence is unmistakable to a lot of people, because it's been used so many times in so many songs.