Walking on flat pavement does not strengthen the climbing muscles. Add enough challenge so that the exercise is aerobic. If you have hiking trails accessible, take your walks there. Try to find a trail that is not smooth. Get your heart and breathing muscles working so they will be strengthened.
Stair climbing is a very good form of exercise.
If you work in a tall building, walk up a floor or two and ride the elevator the rest of the way. Don't over do it and build your strength gradually. Again, the trick is to get your pulse and respiratory rates up. Riding a bicycle is also great. Find a road with some hills and pedal away.