Viv...Ah..the decision between Milos & Naxos - that's tough, but since you have kids and it seems that great beaches and swimming are important - I would choose Naxos, specifically, Agios Prokopios. Naxos is very laid back, family friendly, and is known to have some of the best beaches in Greece. The beach at Agios Prokopios is wonderful. Calm, crystal clear waters with baby powder sand. You can start walking south and before long be in Agios Anna - then farther on down is Plaka beach. This whole area has beach front restaurants and it does have some nightlife but nothing loud or too crazy - for that, go to Naxos Town (Hora) which is only a 15 bus ride away. Buses run back and forth every 1/2 hour until 1:00 AM. We have been to many of the islands and Naxos, with a doubt, has the best beaches. We just spent a week here last month and LOVED it and we don't have kids. There is also a lot to see around the island. It is the biggest of the Cyclades. You can easily take day trips to Mykonos/Delos or Paros/AntiParos. Also, the "minor" Cyclades like Kouffinisia are amazing.
If you by chance end up staying at the Agios Prokopios Hotel, please tell Vaggelis & Effie Hi from Mike & Shannon in Chicago.
Have a great trip. Here are some websites to check out: