As with any exercise, start slow and work your way up. I started swimming more this summer. Just buy a pool pass and do lap swims. To start slow, try the breast-stroke. When you get too tired, flip to your back and do elementary back stroke. Being on your back help you catch your breath, and the stroke is slow enough the effort is more like a walk. One you can breaststroke several laps, start adding laps with a slow freestyle stroke (google around if you need help learning the different strokes). Then as fitness increases, increase the pace and distance of your freestyle and decrease the distance and frequency of your resting strokes. Eventually I got up to the point of doing 200m with a moderate freestyle, follow by a 25m breaststroke. I'd do a total of 1200m (about 45 minutes) and feel like I got a good cardio workout. I swim in the shallow-is (5') side of the pool, so if I have a coughing fit I can stop, stand, and clear out, before continuing on.
The elliptical machine can be used for cardio workout too, but I've never like the motion. You could also consider spinning/cycling or a rowing machine. Again just start slow/moderate and work your way up. It also helps me to do treatments (or at least a MDI of albuterol) before a workout, and then to watch my heartrate. If I get much above 160 bpm, I start to gas out (YMMV), and end up feeling worse after my workout.