I've never rented a car in Europe (just to lay my obvious bias out there), and I wouldn't want one for this trip. Cars are handy for moving from small town to small town, but you're not going to have time for that. Trains will often be much faster than a car, without the hassle of renting and dropping off, the $$$ expense of parking in the big cities, the time needed to find parking at the daytrip destinations (historic districts are usually heavily pedestrianized), or the (slight) risk that you'll be involved in a time-consuming accident of some sort. I assume that your kids will be like most and find the train much more exciting than yet another trip cooped up in a (likely small) car. The trade-off is that you'll have to shepherd your family of four to and from train (or bus) stations. I suggest that you try for accommodations near transportation in each base city.
To keep costs reasonable it will be important to buy your train tickets soon after they go on sale for your travel dates. Right now Renfe is selling through February 28, but early April could pop up at any time. I urge you to check daily. The price you pay should not be more than what currently shows up for Feb 28, and it could be less, since some of the cheapest tickets for Feb 28 have already been sold. The Madrid-Toledo trip is an exception: You need not rush to book that one, because it's a local train and the fare does not vary. I wouldn't wait until day of departure to buy the tickets, though, because Atocha Station in Madrid is large and the ticket lines can be long.
Be sure you're absolutely confident about the date and time you choose, because the cheapest rail tickets are non-changeable/non-refundable. Buy from Renfe (may need to use PayPal since US credit cards often don't work), not for RailEurope, which has a more limited selection and charges substantially more for the same tickets.
Travel Time:
Barcelona-Madrid: 2-1/2 to 3 hr. by train (AVE); 6-3/4 hr. by car (with no stops, per ViaMichelin)
Barcelona-Tarragona: about 30 min. by AVE; 80 min. car
Madrid-Segovia: 27 to 31 min. by ALVIA; 96 min. by car
Madrid-Toledo: 33 min. by AVANT; 68 min. by car
Madrid-Cuenca 55 min. by AVE; about 2 hr. by car
Cuenca-Valencia: under 1 hr. by AVE; about 2-3/4 hr. by car
Valencia-Madrid: under 2 hr. by AVE; just under 4 hr. by car
I included the Cuenca and Valencia legs in there since Valencia was one of your options. It's a city I've never visited, but I but perhaps the science museum(s) and aquarium would interest your children? And Cuenca is a fabulous stop on the way, with a medieval historic district, hanging houses along a ravine, and a couple of small modern-art museums. The AVE station is well outside Cuenca and you'd need to take a taxi or bus into the city. I am not suggesting that you try to do Cuenca and Valencia on a single day trip from Madrid; it would probably make sense to plan one or two nights in Valencia and see Cuenca on the way, after verifying that there's a luggage-storage option in Cuenca at either the AVE station or the separate regular rail station.
With no trip to Cuenca/Valencia, I'd move one of your hotel nights from Madrid to Barcelona. Your first (partial) day in Barcelona will be jetlagged if you're coming off a transatlantic flight, so you'll only have 3 really usable days there. In addition to Tarragona, there are other very good side trips, including Girona (38 min. by AVE, large historic district, wall you can walk on, 2 important cathedrals, and a fine but not large art museum that includes architectural elements from churches that might be of some interest to your children) and Montserrat (good walks if the weather is nice, as well as the monastery). There are exceptions, but most tourists seem to find Barcelona more interesting than Madrid, aside from Madrid's art.