You can swap out the weatherstripping yourself - it's really not that hard. But first you should pinpoint exactly where the draft is coming from, otherwise you'll just be guessing. Try it in the evening with a flashlight: stand outside and have someone inside turn the lights on and off. The light shining through shows you precisely where it's leaking. Or hold a lit candle or lighter around the door frame - the flame will flicker immediately when there's air coming through.
Often it's not the weatherstripping itself, but the frame that's warped. Check if the door is still hanging properly on its hinges or if it's sitting slightly crooked. If the door's 5 years old and only now it's drafty, the house settling could be to blame. Swapping the weatherstripping strip itself isn't a problem - just pull out the old one (sometimes you have to scrape a bit), clean out the groove, and glue a new one in. There are ready-made weatherstripping profiles in different sizes, you just need to know which format you need. Just measure it or bring the old piece with you to the hardware store.
The tricky part: sometimes the misalignment is only in the bottom section, and the top of the door still seals fine. Then you might need a bottom seal - a brush or rubber strip that gets installed at the bottom of the door. That's more involved because the door has to come out, but it's doable. If you're not comfortable with it or the door is really hanging crooked on its hinges, have someone take a look - sometimes an adjustment to the hinges is the quicker fix.