The best way to travel between Paris and London is the Eurostar train, which takes about 2 1/2 hours, plus checking in at least half an hour before departure. Right now they're quoting a $274 one-way total fare for two adults and two teens for a random late-May date. That's a non-refundable fare, and it will go up in the coming weeks and months. Nothing else compares to the Eurostar for speed and convenience. You're wise to fly "open jaw," coming into one city and leaving from the other, which saves time and usually money.
Unlike many posters, you're giving each city enough time to explore and enjoy it. Both offer many choices, depending on your interests. I suggest getting some guidebooks from the library (Rick Steves has excellent guides to both cities, but there re others), and basing your list on what appeals to you most. You could start with the "Explore Europe" tab in this website and click to London and Paris.
Planning a family trip with teens may involve compromises, and perhaps some days when two of you go one way and the other two another. Your kids are old enough to put their own choices into the mix, and they'll enjoy the trip more if they participate in planning it. You might let each family member plan one day for the things they most want to see and do.