It’s really not that hard. You don’t need a special bag like a garment bag. I wear wool dress pants, button down shirts (cotton), tie, belt, etc. I leave a pair of shoes at work because shoes are very bulky and I don’t want to carry those. But for clothes, just fold, then roll them up, and put in a bag/pannier/whatever. I use either a backpack (when I run to work) or a messenger bag (when I bike). I also put the clothes in a plastic bag (thicker plastic) in case it rains (my commuting bags aren’t fully waterproof) before putting it in the messenger or backpack.
I sometimes have to wear a jacket at work so I just leave a few there for when I need one.
So just lay out your clothes and roll them up. They won’t wrinkle for many, many hrs.