another way is to measure your lung capacity. the more acclimated you get to aerobic activity, the more efficient your lungs get at taking in more air. (assuming you're not a smoker, have asthma, etc)
otherwise, i agree that being able to run further, or faster are both good measures. having a stopwatch helps. My wife and I started cycling a lot lately so I added a trip computer to my bike so I know our average speed, trip time, distance, etc. we usually ride the same trail, so that bit is the constant, so i can see if we're improving on any of the other variables, which *should* mean our performance is improving.