From Paris to Omaha/Normandy. Need help!

How much time do you have? The best thing to do is spend a few nights in the area - e.g. in the town of Bayeux. Many people take tours; some people tour the sites on their own. You could do a mix of both: do a tour then re-visit some sites later at your leisure. Renting a car and driving in Normandy is easy and fun - it's beautiful countryside.

You could do a day tour from Paris, but it would be very long day - I would only do it that way if you have no other choice.

I highly recommend at least a one-day van tour of the D-Day beaches. You'll see a lot more and learn a lot more that you will by trying to do it yourself. This means spending at least the night before the tour in Bayeux. You can get there easily enough by train from Paris. If you have more time available, you can visit the WW II Museum in Bayeux, the cathedral and the Bayeux tapestry on your own without needing a car. You can also easily visit the huge Peace Museum in Caen by taking a very short train ride to that city.

If you're willing to rent a car, you'll have more flexibility to range away from Caen-Bayeux without being dependent on train and bus schedules, but I would still recommend a D-Day tour.

Trains to Bayeux/Caen leave often from Gare St. Lazare in Paris. There's a ticket office for Normandy bound trains in the corner of that station, or you can buy online in advance. Price is around 20 euros. You can look up more at trainline.eu

We took the train from Paris (changed trains in Caen) and on to Bayeux. (not expensive at all.)
We were able to walk to our hotel from the train station (in the center of Bayeux.)
We took an all day tour of the beaches of Normandy, Cemeteries, and small towns significant to D-Day. It was wonderful to be at the American Cemetery at 4:00PM when they lower the flags and play taps.

We also did a 1 day tour to Mont St. Michel.
We booked our tours through Viator.com (2 different tours- each 1 day) and it was perfect. Our tours through Viator were actually conducted by Overlord Tours.
We had the same guide each day and he was wonderful. It added so much to have a great guide to take us on a very meaningful tour of Normandy, have a nice lunch, visit several small towns in the area.
Next day was Mont St. Michel - another all day tour, including a great lunch and we walked to the top of the Mont and learned so much about the island.

You can also easily visit the huge Peace Museum in Caen by taking a very short train ride to that city.

This is not at all true. The train station in Caen is the other side of town, and is an hour walk from the Caen "Peace" Memorial. You would need a taxi, Uber, etc. Personally, I don't think the Caen Memorial is worth the time. even if you have car. IMHO, there are many better museums/sights in the Normandy area.

Best/cheapest route to the general area is a train to Bayeux or Caen from Gare Saint Lazare in Paris, but realize that this is a huge area. You will need a car or a tour to get you from Bayeux to the beaches, cemetery, or any other worthwhile WWII sights.

On the other hand, the WW2 buff could stay in Caen and take a day tour from the peace museum (yes, it is on the outskirts of the city.) Either way there is more than enough for a full day guided tour, not cheap but far more informative than DIY. As well, the British Commonwealth armies have significant locations of their own, which some tours includet.
Repair work on the train line from Paris to Normandy has been ongoing for several years (or so it seems) but sometimes bus service will be provided around the area of work. And sometimes the delays have been confined to weekends.

PharmerPhil is right; I should have mentioned that the Peace Museum is nowhere near the Caen train station. However, there is decent bus service (3 euros roundtrip as of last summer) from central Caen to the museum. Service may be less frequent on Sundays.

Spend a couple days in Bayeux. Book the D-Day tour via Bayeux Tours for one day. It will take you on a guided tour of several sites & the German and American cemeteries (including the evening flag ceremony...amazing). Another day, walk to the Bayuex D-Day Museum in Bayeux (small but very good) and then up the street to the British Cemetery. Stop in the park Place Charles de Gaulle to take in the spot where de Gaulle gave his speech.