Trips outside of Paris?

I think it depends on your interests. I've been to Burgundy and am currently planning a trip to Normandy and the Loire Valley. If you like wine and exploring cute villages, you will love Burgundy. I stayed in Beaune, and greatly enjoyed exploring the town (especially the market, which I think sets up once or twice a week) and touring the Hospices de Beaune. Then I also did some wine touring. You could do this with or without a car. With a car, you could drive to different wineries and villages on your own. Without a car, it's a pretty easy train ride from Paris (I believe one change, no more than a couple hours altogether), and you could sign up for a tour to explore wineries and the countryside. I did a bike & wine tour through this company (http://burgundybiketour.com/) and loved it.

If touring extravagant chateaux is more up your alley, you'll enjoy the Loire Valley. And if WWII history interests you, Normandy would be a great option; there are lots of other things to see in Normandy as well, so do some research. For both of these options, I think you'd be best off with a car, though it's not impossible to do without one (again, combination of trains plus using tours to get to some of the places you'd like to see).

For all of these options, I'd suggest two nights. And I would definitely book a hotel in advance. It's unlikely you'd be unable to get a room in early April, but who wants to waste time trying to find a hotel room while on vacation?

I would suggest if you do that, that you proceed to your destination on arrival e.g. arrive in the morning and take a train to the region early that afternoon. That way in such a short trip you are not wasting the second to last day rushing back to Paris. Frankly with only 7 nights/6 days I would spend it in Paris. That really barely scratches the surface. If you must have variety take a day trip in the region e.g. small towns like Senlis, Moret sur Loing, Crecy la Chapelle, Auvers sur Oise, Provins

or chateau like Fontainebleau, Versailles, Vaux le Victomte or small cities like Rouen or Reims.

If you do decide to do this one possibility would be to train to Dijon or Beaune spend the night and visit the city and then pick up a car and spend a couple of nights there and drop the car at Orly on your way into Paris.

https://janettravels.wordpress.com/category/burgundy/
Normandy is also a great place to stay and you could take the train to Bayeux, stay 2 nights or 3 at Hotel Churchill which is across the parking lot from the Bayeux tapestry and then take their van tour the next day to Mont St Michel and a half or full day tour of the American beaches the next day and then train back to Paris and finish there. You want to be in the town of departure at the end.
https://janettravels.wordpress.com/category/normandy/
One week is a perfect introduction to Paris and too rushed for another region IMHO. If you decide to do another region get train tickets 3 mos ahead where you can get PREMs to print at home. It is 15 Euro to and from Bayeux about a third what you will pay last minute. And definitely line up your hotel well in advance.

I agree with Janet about spending your whole week in Paris, especially if you've never been there before. There will be more than enough to keep you busy and happy, plus time to sit in parks and cafes and enjoy the scene, plus time for one or two day trips elsewhere if you want. Janet listed several options, I'd add Chartres, about an hour each way by train. And her advice about being in Paris the night before your flight home is also very good. If it's a morning flight, consider spending the night at one of the airport hotels. Don't be racing back to CDG from Burgundy or the Loire or Normandy or somewhere else the same day you're flying -- unless you can afford to miss the flight and buy another ticket on the spot!

Early April will likely be cooler and wetter than later in the spring (duh...), and if you spend nights outside Paris you should have hotel options without reserving in advance. But you might think about how much time you want to spend "on the ground" looking for lodgings, vs. knowing where you'll stay before you get there. April weather would be another reason for me to prefer to stay in Paris, where there are lots of indoor options and good transportation. If you stay in Paris you won't need a car at all, likewise if you take a day trip to the places mentioned so far. But if you do spend multiple days in another region, then a car would help you make the best use of your time there.

I too would urge you to spend all 7 nights in Paris, giving you 6 full days.
Day 0. Arrive Paris
Day 1. Notre Dame, Sacre Couer, Eiffel Tower
Days 2-5. Four day Paris Museum Pass (including one day to Versailles)
Day 6. Chartres, or Giverny+Rouen, or Reims
Day 7. Depart Paris

I do not think you can go wrong, actually. 7 days in Paris, with a day trip or two would be great, and there's plenty to do when it rains (which it will). But all of your other ideas are feasible too, and any could be delightful as long as you are ready for wet weather.

So what interest you the most? In every case, I'd suggest taking a train to the region from Paris, and then renting a car for the few days you are there (wherever there is). That time of year you can get away with last minute bookings, if you are not too fussy about where you stay.

One thing about these short itineraries is the "lost" day, where you return to Paris from, say, Burgundy the afternoon before you have to leave, and then spend the last 24 hours of your trip basically waiting to get on the plane home (with, admittedly, the possibility of a nice meal and a romantic walk). It's an anticlimactic way to end an epic trip.

The solution, which is far from perfect, is to flip the order and see Burgundy/Normandy/Loire first, Paris second. It saves you from a hotel change and uses your time better, but of course it makes for an extra long travel day.

Add my vote for all seven days in Paris. You won't run out of things to do, and your trip planning will be much simplified.

Paris all seven days. Be sure to visit Ste Chapelle near Notre Dame. Ste Chapelle's stained glass is spectacular - best I've seen.