Rhodes Island has quite a few picturesque locations that you won't want to miss either for your holiday record or for your artistic satisfaction! As a professional landscape / travel / nature photographer I will try to give you an insight on the locations, times and requirements to help you prepare.
I will try to satisfy the casual, amateur and professional photographer, if you have any specific questions / suggestions you may send me an email and I will do my best to follow up.
Rhodes photographic interest may be categorized in Landscape, Nature, Monument (Historical), Sports, Underwater themes.

Useful equipment, Tips:

Landscape Photography in Rhodes Island
Rhodes Island is offering good Landscape photography locations, it has very nice summer and autumn sunsets. Usually with a beach setting, but also with panoramic views from high up, like the areas of Monte Smith and Kritinia. Other landscapes with rocky coastline, crystal clear turquoise waters are also good options, especially with use of polarizer filters. You will need to print the sunrise / sunset schedule for the dates of your visit to be better prepared. Note that in beach settings, the sand may be an issue for your tripod, in some cases its very thin, in others it is blown by the wind to all your equipment, and it may cause permanent damage to the tripod joints.
The best seasons for photography due to the clear atmosphere is on Autumn from October, and Spring at about March / April. Summer has thick atmosphere, which in turn creates very nice and colorful sunsets and sunrises.

Nature Photography in Rhodes Island
Nature photography in Rhodes Island, can be targeted to forests, insects, wild flowers, some wild animals, birds, reptiles, fish.
Wild animals in rhodes are mainly Deers which can be found in south Rhodes at the forrests but are very hard to find, you will probably need the help of local villagers to point out places where the animals drink water or passages.
Rhodes has a large number of lizzards, especially in the moat of Old town and the castle walls, the Ancient Stadium of Rhodes, and many other rocky locations.
In terms of birds and Bird Watching, Rhodes is a migratory passage between Africa, Asia and Europe, so it is quite rich in bird life at various seasons especially Spring time.
Very good locations are the South Rhodes at Apolakia (near the damn is a good location due to the lake), Gadouras River, Koskinou, Kalithea, Kremasti bridge, Ialyssos area as well as the Monte Smith hill and the hills close to the coast of Rhodes Town. Also good locations are the Profitis Ilias mountain as well as Attaviros mountain (which can be visited by car up to its top). Birds you will find easily are: Eagles (few), Hawks (more popular), Peregrine falcons, Owls, Vultures, Ducks and Gooses, Doves, Ravens, Sparrows etc..
The Aquarium of Rhodes has a good collection of fish, which can be photographed with a tripod (no flash allowed - it is not useful anyway because it creates reflections on the fish tanks).
Rodini park is a nice natural site with a river, gooses and ducks, turtles and peacock birds, as well as other small birds and hawks.

Monument / Historical / Landmark Photography in Rhodes Island
Rhodes has a long history from Ancient times to Medieval, with many historical monuments, all around the island. The famous Old Town of Rhodes, in which you may spend a full day walking and taking photos in its narrow streets and various monuments and museums, the 3 ancient city-states of Rhodes, Lindos, Kamiros and Ialyssos (Ialyssos unfortunately doesnt have many monuments to be visited like Lindos and Kamiros), Rhodes Town with its Ancient Stadium and the Apollo temple, the Mandraki area with the lighthouse and various other locations spread through the town.

Sports Photography in Rhodes Island
Sports photography is basically water-sports photography since the most spectacular and popular sports are the water-sports, especially Kite surfing and Windsurfing. Thousands of athletes from all levels are visiting Rhodes for its ideal weather conditions that almost guarantee wind and waves.
In addition, many windsurfing and kitesurfing schools offer professional equipment rental, allowing more and more enthusiasts to visit the island for their sport.
Ixia and Ialyssos beaches host many athletes, from late morning till late afternoon, for windsurfing and kitesurfing, they usually are close to the beach so you dont need a huge lens, a 70-200 or a 300 will do a very good job, just keep in mind that the wind is blowing towards your lens, and it usually has salty water spray, which will require you to thoroughly clean you equipment after the shooting...
In Prasonisi area which is a heaven for windsurfing and kitesurfing, you will have also the thin sand to fight off, so a plastic protective case of your camera would be of great help!!
There have been many national and international competitions in the past in Windsurfing and Kitesurfing in Rhodes, so if you visit at a date of a competition, you will surely have a chance to photograph the top athletes in the world.
There is national powerboat championship held every summer as well as sailing races that pass through Rhodes.
There are more watersports available, but not at professional level.
In terms of other non water-sports, Rhodes is at the 1st national basket category and the 2nd national football category, but these games are held during the autumn, winter and spring.
Motorsports include some local gokart races, an annual car race (not during the summer though)

Underwater Photography in Rhodes Island
Rhodes has a very good underwater environment with crystal clear waters, caves, sea life, ship wrecks. There are many diving / scuba schools that regularly make diving trips at various locations, the visitor can learn and get a diving diploma or just rent the equipment for diving, and follow the group. Most diving schools depart from Mandraki Harbor, and you can talk to them at the evening, someone is usually outside the school boat to answer questions and book a lesson or rental.
In Greece a diver has to be over 14 years old and up to 60 years old.
Popular scuba diving locations include: Kalithea Spa where the sea-life and the underwater landscape are ideal for beginners, Ladiko beach, Lindos, Kolymbia, Pefkos, Plimiri (cargo ship wreck)
Even though there has been a recent opening of more sites to dive with underwater diving equipment, you need to talk to the diving schools directly to find out the legally available locations for diving, and make sure you dont dive without a local guide because it is illegal to dive in many locations due to the archaeological underwater sites, and the Greek law is very strict on that!!
Snorkeling on the other hand is also very popular and you can do it anywhere, with top locations for underwater landscape and sea life photography while snorkeling (relatively shallow waters): Antony Quinn bay, Lindos St Paul's bay, Kalithea Spa and surrounding rocky coast areas.
SAFETY TIP: When you snorkel, make sure you have company, and if you are at deserted locations, it is mandatory to have an identification balloon to warn speedboats. Try to stay close to the coast and outside the speedboat balloons.

For more samples, check out my professional stock photography collections of Rhodes Island at IStockPhoto and GreecePhotoBank