Out of the 3 locations, which do you recommend for lodging and access to nice beaches and ease of transportation around island.
The answer may change depending on what time of year you are planning and if you are renting a car. Certainly St george beach gives you the best access to a beach the town and public transportation around the island.
The other two beaches are a half kilometer apart so are essentially similar. Both have good beaches and villages for your needs, To use public transportation around the island you will have to take the bus back to Naxos town. If yo have a car then it doesn't really matter where you stay.
Best beach of them all is Plaka beach
St George Beach Naxos http://www.flickr.com/photos/stanbr54/sets/72157632113170279/
Plaka beach tavernas hotels https://www.flickr.com/photos/stanbr54/sets/72157655106193251
Aggia Anna and Ag Prokipos beach areas https://www.flickr.com/photos/stanbr54/albums/72157656297072028
Naxos Town http://www.flickr.com/photos/stanbr54/sets/72157632094558042/
Trip around Naxos http://www.flickr.com/photos/stanbr54/sets/72157634605629689/
We stayed at the Galaxy Hotel on Naxos in 2015. https://www.hotel-galaxy.com. It is very nice, has a great breakfast, is steps from the St. George Beach, as well as being about a 5 minute walk into Naxos Town. The bus is very close and we visited the other two beaches. We loved Greece so much we are planning another trip in two years, this time to Crete (we think).
mid July :(
Mid July is high season so St George beach will be fairly crowded but if you walk to the far left facing the sea you will find a long stretch of beach with dunes. There is no infrastructure out there so it isn't crowded. while Plaka and Prokopius will have some uncrowded sections. Local bus will be every half hour so you can get to Naxos town easily.
Mid-July is High Season so most anywhere in Greece will have crowds, some more, some less, even on the smaller, less touristed islands.
We stayed in St. George Beach just outside the Naxos Waterfront. A basic, but clean, safe and centrally located studio. Lovely views of the beach, bay and distant ocean. A short walk to the waterfront and Old Town and an even shorter walk to a small town square with everything you need for a night out but not overly touristy.
Stanbr's photos are right-on and will give you an idea of what to expect. We stayed in October so the beach was basically deserted and we had it to ourselves. The weather still mild and water still warm.
I'm sure July will be evern warmer and have more people but should provide you with a nice Greek Island experience.