Sardinia for beginners

I always stayed in Northern Sardinia.
I like Santa Teresa di Gallura. From their harbor they have numerous boat excursions to the Maddalena Archipelago. I recommend it, I am sure the kids will appreciate as well. The excursions are weather dependent and don't run with high winds, which are common there, so try to do it early during your stay, so that if canceled you have a chance to try again the next day or so.
One year I stayed at Capo Testa, near Santa Teresa, at a budget hotel resort called Hotel Mirage. The hotel was so so, but very close to Cala Luna beach, which was very nice.

You can google Sardinia for ideas, or you can also look at the official tourism board website below:
https://www.sardegnaturismo.it/

I also really like Santa Teresa di Gallura! However, I would be cautious about beach time at the time of year you are planning to go. We've only been to Sardinia once and it was the last week of summer (mid September). Most of the week was lovely and perfect beach weather, but by the end of the week, the weather had really started to change- on our penultimate day we wound up sitting on the beach wearing jumpers and the day we left was a huge rainstorm. Once you get into late September the weather is a lot less predictable. I remember looking at the forecast for the week after we left and knowing we wouldn't have been able to go to the beach. If the beach is one of your main draws, I would try to move your trip up a bit earlier. Also things that are seasonal (like boat trips) will start to close at the end of September, so you'll want to check that.

Other things to mention: Alghero has a beautiful historic old town and a very nice beach. The public transport isn't very good so you'll probably want to rent a car. If you like history, you can visit some of the Nuraghe: https://en.wikipedia.org/wiki/Nuraghe

Good luck with your planning.

It’s not beach season, and though you could luck out and have great beach weather, I would make a plan where you have alternatives if it’s not good swimming weather (wind can be a factor even if weather has not turned). Though nature is the big draw for me, Sardinia does have some historic interest and the archaeology is fascinating.
Lonely Planet and DK both have good guides. With five days, I imagine you will stick to two stops max (I suggest one), so first read up on the three airport cities/areas (unless you plan to arrive by ferry) and one should rise to the top in terms of your interest. Once you have narrowed it down a but, happy to chime in with more suggestions specific to your plan.
Things to be mindful of: your kids’ hiking ability, any motion sickness so drives can be selected carefully

Knowing what I know so far, if it were my trip, I’d fly to Alghero and explore that area, where you can dine in Sassari, explore Alghero’s historic core, hike and do boat trips and go to beaches in the area, go to archaeological sites, do a day trip to pretty Bosa.

We (family of four) visited Sardinia in the first week of September, two years ago. We had great weather, and were able to spend time on the beach everyday.

We spent all of our time on the northwest coast. We enjoyed Alghero, and you can get a boat ride to Neptune's Grotto. Bosa was fantastic, but might be hard to work into you schedule. Not so much a day trip as a place to stay overnight. Further south, I liked Oristano, which could be paired with a visit to San Giovanni di Sinis where you can go to the beach or visit the archaeological sites.

As mentioned, a car is a very handy thing to have during your stay. We came in on the ferry from Barcelona with our car.

You might also think about staying at an agriturismo which might have activities that kids might like.

So long as you don't have unpleasant rainy weather, you should have an enjoyable trip where you go.

Hotel da Cecco is a cozy family-friendly hotel in Santa Teresa di Gallura. The location is great and you can see the sea from your room. The hosts are very kind,

We stayed in that same hotel (da Cecco) and were also very happy with it.