I had a lovage plant for about ten years. When we moved, it had reached a dizzying height of 6.5 feet. It was a beauty.
Harvest young leaves and treat them as a herb. They go well with leeks, celery, onion, parsley, potato, poultry (try roasting a chicken set on a bed of lovage). It is nice torn up in salads. Taste as you go because it is quite robust.
I never did cook the stems but I imagine young tender ones would work best.
good luck.