Depends on your budget, if you have money to rent a car then do it. Almost every time at all Greek islands the price to rent a car comes from 30-40 euros to 60-70 euros per day, depending on the car. As far as you are going on October then, fortunately for you, you will be lucky on prices because of the low traffic of tourists at that period.
Otherwise if you want to use public transport then do it, tickets on buses of the islands cost about 3 euros (for one way road) but going among the island so you could use them to explore the place and of course you could ask them for suggestions, they know better!
Mind if you would like to take a taxi then to ask before for the cost of the road, otherwise someone will want to treat you and will ask you much more money. So be careful to avoid that.
My suggestion is if you can wake up at morning and you can end you exploration almost at 20:00-21:00 hours (after that hours you probably will have to wait long and maybe buses don't come,so better to be sure), then use public transport.
If you want to be more relax and you can afford rent a car.
Don't worry all buses goes everywhere on the island so that is not the problem.Depends on your money and how you want to explore the place.
At any case I wish you "Καλά να περάσεις στην Ρόδο", it is pronounced like "Kala na perasis stin Rodo (the name is Rodos in greek not Rhodes, that's English)" and means "Have fun at Rodos".