judy,
To begin with I assume you're referring to Gare Saint-Lazare rail station? As it looks like you'll be going directly to the station after a long international flight, Taxi is certainly going to be the easiest method, albeit one of the most expensive. Hauling luggage on the Metro or RER after a long flight will not be the most pleasant experience (I've done that).
Despite your previous bad experience with a tour, I would highly recommend taking one of the excellent tours in that area, as you'll learn far more about the history and it will be a much more interesting and rewarding experience (I've done both so can compare them). Even with a car, the battle sites are spread out along about a 50 mile front and extend some distance inland, so even with a car it could take awhile, especially if you get lost.
One of the tour companies that seems to get consistently good ratings is Overlord Tours. They offer both half-day and full-day tours, so would provide the best flexibility. The guides know the shortest distance between historic sights so that's a very efficient way to see as much as possible in a limited time. Since you'll have a car, one other really practical suggestion would be to take a tour with Chris at Visit DDay Tours as he will travel with you in the car. The tour could be tailored to your specific interests.
I'll leave the question on wardrobe to the ladies on the forum, but Bermuda shorts wouldn't be my choice. It can often be windy close to the coast and depending on the weather at time time, shorts may be a bit "chilly". From a practical point-of-view, zip-off pants would probably be a good solution.
One final point - as you'll be using a rental car, be sure that you have an International Driver's Permit or a certified copy of your D.L. in French. Those are compulsory for driving in France.