Mykonos is known as a party island. It is going to be busy and more expensive than other islands, but you are still in Greece. Expensive is a relative term. It is a beautiful island with much to see just in town. The bus from the new harbor to town is kind of a joke; it is only 200 yards, an easy walk. Visit in the summer time for the best results. All the restaurants and shops will be open and ready for you. Off season can be limiting although we had a great time on Mykonos and Delos.
Naxos is a larger island and not as touristy, but still has great beaches. Plaka Beach is well known, but out of season none of the restaurants were open. Again, visit islands in the summer. We rent an ATV and had a great time. The beach was mostly empty while we were there. Temples and the best graffiti in Greece are on Naxos.
Watch the video for great pictures and advice on traveling in the greek islands.
Video link: Mykonos and Naxos