It takes about 2 1/2 hours to go by train from Basel to Lauterbrunnen.
Your basic itinerary sounds good for a one week stay. You won't need a car for any of it. Ken mentioned that a car would simply gather dust in a [Lauterbrunnen or Stechelberg] parking lot if you stayed in Mürren, where no cars are allowed. The same is true in most of the other mountain villages and trail heads in that area. It's all well serviced by trains, cog rail trains, cable cars, gondolas, and buses. And if you take the train to one of the big lakes nearby, those are serviced by ships/ferries.
Luzern is very walkable, but also well serviced by buses, trains, and ships/ferries. And the mountains nearby are serviced by cable cars and cog rail trains, as well as ships/ferries, buses or trains to get to them. (Although I haven't been there in April, which is off-season, and transportation schedules might be reduced then.)
As Jazz+Travels pointed out, you can't fly from Luzern.
But since Luzern is only about 40 miles from the Zurich airport, you should plan to fly home from Zurich. Trains can take you right into the Zurich airport, so no need to spend your last night in Zurich.
As you can see from the SBB transit link that acraven posted for you, starting at 7:10 a.m., each of the trains that leave the Luzern station at 10 minutes after the hour bound for the Zurich airport get you there in 1 hour 3 minutes with no change of trains. That's a nice convenience when you have luggage. There are other trains, too, but you'd have to change trains once to get to the airport.