Archaeological Park Neapolis- This is a wonderful park with an ancient Greek theater, Roman amphitheater, a famous cave named by Caravaggio and more. It's all a short walk around a peaceful park where you can see many ancient ruins, and you even get to walk on the seats of the Greek theater.
Museo Aretuseo dei Pupi- A small but delightful museum with finely crafted puppets and scenery.
Ortigia Street Market- While I was there it happened every day, and was a wonderful place to people watch and look at all of the amazing fresh seafood and produce. It is also where we finally found fresh ricci di mare (sea urchin), and it was delicious!
A Putia delle Cose Buone- This is a lovely spot where I had my first meal in Syracuse/Ortigia. The decor is artsy, the staff is friendly, and the food and wine is delicious and fresh. I loved dining here. It is a small place, so I recommend you go early to get a table.
Paninoteca da Antonio e Daniele- An amazing food-truck where you can eat a horse sandwich if you choose, or plenty of other options too. I like to try things I can't always have, so I ate the horse sandwich with cheese, and it was fantastic. If I had to compare it to something I would say it is similar to a Philly cheesesteak. It opens at 17:30 until late.
Fratelli Burgio- Meat, cheese, wine and all things delicious are at this store. The gentleman behind the counter is super helpful and even gives you samples to try before you purchase your items. I bought local spicy salami, local Porcino, and local wine; everything was fantastic. They will package it for you, so that you can bring it home too. Wonderful place to shop for the best items Sicily has to offer, it's food.
Caseificio Borderi- Right next door to Fratelli Burgio is the famous sandwich maker at Caseficio Borderi, and you can believe the hype: the sandwich is heavenly. It is a famous place so you are going to have to wait, but while you wait they give you samples of cheese and even wine. They chat with you and keep the whole crowd in a happy mood. I myself found it mesmerizing to watch the father make the sandwich at this family-run establishment. I like to say the sandwich actually saved my life, because I purchased it shortly before taking a horrible ferry ride, and I think without the sandwich I just might have perished. Go as early as you can to beat the long line (it's worth the wait). You only need to know how many sandwiches you want, so don't worry if you don't speak Italian. Yes, you should get the cheese they offer you.
Il Blue- I came here for a glass of wine, but I ended up dining here too and I'm glad I did. It's a family-run/husband/wife team restaurant. I had the Fish tartar and a wonderful local pistachio pasta dish. The seafood was incredibly fresh and the service was great. I was here in winter, so I did not get to enjoy the sea, but the decor is tasteful with black and white photography on the red walls, and candlelight.
Mamma Iabica- A cute cafe that sells speciality items like local wine and biscotti. I enjoyed a cappuccino, and bought some wine to take with me. The service was pleasant,and the decor is vintage.
I highly recommend you stay on Ortigia by the sea. It is a gorgeous location, and you are close to the sites and all of the best food/restaurants. It is only about a 25 minute walk from the train station, and a pretty straight easy walk too, or you can take a cab that will cost you about 10 euro for an 8 minute ride.
Always ask how much the cab is going to cost before you accept the ride, because they way overcharge for cab service on the entire island of Sicily, in my experience. Way, way overcharge, and they never have meters.