Question Best Way to Travel from Lucerne to Fussen to Salzburg?

I'm planning a trip and need some advice on the best way to travel from Lucerne, Switzerland to Fussen, Germany, and then on to Salzburg, Austria. We're considering public transportation options like trains and buses but are open to other suggestions if they're more efficient or scenic. Has anyone done this route before? Any tips on schedules, costs, or must-see stops along the way would be incredibly helpful. Looking forward to hearing your experiences!