I weigh 225, so not fast at climbing IRL or on Zwift but I actually enjoy it - more feet climbed definitely translates to me being faster on both the next climb and on flat rides. I tend to seek out hillier rides outdoors.
The obvious advice is that the more you do climbs the better you will feel after the next climb.
I also found high intensity interval workouts to be shorter workouts that translate into improving my climbing speed. Tabata intervals are really good - there are several in the standard Zwift workouts. There are also various group workout rides that feature intervals.
There are routes on Zwift that are fairly short but mix a few sprints with a few short but steep climbs - the London Loop, the NY Park Perimeter and others aren't really interval rides but give me a good mix of high power bursts.