We can't give you the best advice without more complete information -- you may have shared this on previous threads, but should not count on contributors researching your past messages, we do this gratis, and we have busy lives too. LIke for instance -- when are you going? Do you already hve accommodations booked and where? How many days do you have on Naxos? Do you have a map of Naxos, or have you looked on Google Maps? Sounds like you haven't or you wouldn't have asked about how to go between beaches. Some observations:
Naxos has an excellent bus network of course frequency goes down markedly i nthe off season. There's a bus that goes fro Naxos town and runs all along the S. West BEaches in order, Prokopios, Anna, Plaka (my fave), some go as far as Mikra VIgli. Bus stops several places along each beach. During July-august, every 30 minutes and until midnight; otherwise hourly, til about 11pm. If you want to rent a quad, I think u need a driver license and an IDP (international driving permit -- get at AAA in 20 mins for $25). Road to beaches & along is flat, no problem; challenging up in the hills.
BUS TOURS? I don't like those by bus company, or by Zas -- I like https://naxostours.net/en/bus-tour-around-the-island/ that gives you 90 minutes up at Appollonas where you can use the time for a swim. These do have a guide who can talk about the ruins. However with a car rental you can pick & choose... You don't have to stay in a village, for example, just drive thru FILOTI (or stop beneath the huge tree in its center, for a coffee & watch geezer play dominoes) ... stop in CHALKI just to visit the Kitron Distillery (and u can walk thru a wooded path to an abandoned church, very evocative)and a marvelous craft/design shop/bread & Olives?? APIRANTHOS is atmospheric because it's entirely PAVED in marble... and has a tiny archeological museum with one amazing standout item -- a stone-age ROCK with 5,000 year-old scratched drawing of stick-figures DANCING (I kid you not). Then going west toward Melanes, the teeny-tiny (small as a 1920s garage) Oldest Church in Cyclades, made of fieldstone. All of these are visited by a bus tour -- but the advantage of a car is that you aren't tied to a group, and you can move on quickly, once you've seen what you want, no waiting upon others.
Moutsouna is basically a deserted port for the long-over export of emery... I hear there's one taverna around ... beware of roads leading to/from. I haven't been to Tripodes (as Vivlos is also known) but it's near Plaka beach ... again, you can get confused on the dirt back-roads, but it's all part of the adventure.
Finally, 2km outside of MELANES, is an abandoned Monastery -- built in 1600, empty 250 years, beautiful ruins - https://www.greeka.com/cyclades/naxos/sightseeing/kalamitsia-jesuit-monastery/ Also, right by the road near Melanes (stop in a store and ask around) is a fascinating "Marble Mill" where a huge saw slowly slowly slowly cuts thre enormous slabs of marble, and in an adjoining building, cuts & polishes various shapes. Lying in the grass by the roadside -- dozens of discarded 3 x 3" marble squares (now on my coffee table as drink-mats).
SO you see, a ton of options; I've been to Naxos 13 times in 20 years, and still make discoveries, but I DO read guides and each time seek out something new; for instance in the courtyard of Cathedral, right up from port, there's a low shed... walk in and WOW a Mycenaean Graveyard! They found it when re-doing the pavement, so they just covered it with glass blocks & you can walk over & stare down at grave ritual items 3,500 years old. Lots to discover.