Congratulations on your wedding.
You should start to think in Euros so everyone understands what your budget really is.
There will be lots of choices of hotels in your range but it is starting to get late for new bookings. I suggest you go to booking dot com and put in your dates and you will bet availability price and reviews. This will help you narrow down your choices.
Where to stay. Personally I prefer to stay at St George beach. It is part of Naxos town so you get the beach holiday plus all the activity of Naxos town. St George is just far enough away from town that it is quiet at night for sleeping. Greece can be very noisy at night.
St George Beach Naxos http://www.flickr.com/photos/stanbr54/sets/72157632113170279/
St George is a good beach but there are better ones out on the west coast about 7km from Naxos town. These include Agia Anna. A small narrow beach with hotels right up to the sand. Ag Procopios a long crescent beach with a village.
Aggia Anna and Ag Prokipos beach areas https://www.flickr.com/photos/stanbr54/albums/72157656297072028
Lastly Plaka my favourite. It is about 3km long with hotels and tavernas across the shore road. The further out you go the less amenities there are but the more free space there is on the beach The further out you go the skimpier the bathing costumes as well.
Plaka beach tavernas hotels https://www.flickr.com/photos/stanbr54/sets/72157655106193251
If you stay at St George there is no need of a car. Everything is within walking distance, including the bus stop to the west coast beaches. You will want to rent a car for a day to visit Naxos mountain villages and interior. There are lots of rental agencies at St George just book a car the night before you want to head out. You will need to know how to drive a stick shift.
The beaches are well serviced by local bus which is inexpensive and frequent so you don't really need a car until you want to explore the island. There is a rental agency in Procopios. You hotel can also probably arrange a rental for you.
Here is what you will see on Naxos
Naxos Town http://www.flickr.com/photos/stanbr54/sets/72157632094558042/
Trip around Naxos http://www.flickr.com/photos/stanbr54/sets/72157634605629689/