Great place @Mummabear04
I went in early July. With young teens. I'm mountain leader, my kids have grown up climbing Munro's.
There's a lot of mountain walks and trails. They are well marked, busy and easy to navigate. They are pretty big in height gain - if your kids are used to Lakes or Scotland they will be fine, if not they might find the bigger walks pretty brutal. There's scrambling in places, but they use chains and chip out handholds, even install ladders.
It was/is properly big mountains, and all the adventures that brings. I'm not sure mine would have enjoyed it as much if they had been the age yours are.
We did Giewant, Swinica, Rysy (big day!), Kresanica, Kosi Wierch. Also the hike up to Morsi Oki (sooo busy).
We didn't stay in huts, but I've friends who have and found them excellent, easily as good as those in the Alps.
We did have a hire car which was useful, stayed at a nice self-catering place with a great view.
The day we went to Giewant the rangers were warning everyone that was going up that the bears and wolves were both in the valley that day and young families should not go up, nor any dogs. It adds a frisson to things...
There's lots to do in town, including a great swimming pool and lots of shopping, plus the oddity that is Gubalowka, some lovely lakes etc.
Read up on mountain rescue - basically the Slovak's fund a rescue service, the Polish expect you to pay. Make sure you have the insurance to cover mountain rescue and the higher altitude.
Things I found difficult:
- the cable car queues. We gave up one day after arriving at 9am and still waiting at 1.30pm, having moved a few hundred metres with another few hundred metres to go. Next day we were at queue at 7am and were on almost the first lift at I think it was 7.30am.
- the afternoon thunderstorms were very powerful. It was amazing how many people were still heading up the hills into the face of really big storms....and I would say the hill tops had storms on 1/3 of afternoons.
- how busy and slow a few hills were. Giewant was just crammed, with really really slow queue to scramble down from the summit. It led to us being caught in a big thunderstorm - we managed to find a large rock to cram under and sat it out until we got too cold and the lightning eased...
- grading of walks was generally over graded, but we had one 'walk' that included a 20m vertical ladder up a rock face...! Do research before and ask locally...
I would also say that one of the best family walking places was the Picos de Europa (1 week) and the Basque country (1 week). Great mountains, good walking and trails, very quiet and more rural for the most part. Amazing wildlife (chamois running past us, vultures flying within a few metres with 2.5/3m wingspan, butterflies galore, crazy lizards...). The weather was more reliable than Tatra, but still some afternoon storms and a 'summit before the sun gets to it's peak' approach to life (no lie-ins...). I stayed in Coolio which was high enough to loose some of the Spanish heat, close enough to Pores, Bulnes and more for town and shops, access to walks etc. The scenery I would say was even more stunning, and many of the best walks were easier miners trails or with parking high up the hill. Basque I stayed near the coast near Ondarroa, but headed to hills upto an hour's drive away.