Only bring comfortable shoes, preferably SNEAKERS, there is a shit ton of walking. And that's the easy part. It is all steep, cobble stone steps that you can barely navigate. Now don't let me scare you, its not terrential but there is no room for Louboutins in the suitcase for the Greek Islands. If you bring, them, they will be a waste of space. Find the cutest sneakers you can get your hands on (and feet in) and pack those!
Make sure you are somewhere seaside for the incredibly iconic Santorini Sunset. To say it is breathtaking is an understatement. Even better, take a catamaran boat ride around the island to have the best seats for the sunset. They typically last 5 hours and serve up some souvlakia and drinks. Bring your swimsuit and dive right into the beautiful blue ocean. It is an unforgettable experience. You definitely want to book this in advance, you will get a better price. They have morning and afternoon sessions, so you can pick which works for you, but I definitely recommend the sunset cruise.
Unpopular reality: Santorini is not the greatest beach destination in Greece. I cannot lie to you, Greece has so many stunning crystal clear blue beaches, but Santorini isn't one of them. There are many beaches but the water is dark and uninviting. If you are there for a few days, definitely visit the Red Beach and Perivolo Beach.
A must see in Santorini is Amoudi Bay. An absolutely beautiful spot in Imerovili where you can enjoy tavern style Greek food and watch the sunset. They have the best food at these taverns where you just want to order all of the appetizers on the menu and go to town. This is the place where you see the hanging octopus and I'm telling you: go there hungry!
A few other recommendations for the island in and around Firostefani is the Volkan for dinner and a movie. Yes they have an outdoor theater in the middle of town but you view with headphones not to disturb or be disturbed. Definitely a unique experience.
Make sure you visit the main town of Fira, where you will spot the famous donkeys and go shopping for souvenirs. The shops are open late and that makes the evening more pleasurable. There are plenty of after hours bars and night spots in this town as well.
A car rental is not necessary but recommended and usually not too expensive. You can also get around the island by bus (the least expensive option) or TAXI. There is also the less expensive rental option of the mopeds.