Ideas for Sicily

http://www.holiday-weather.com/taormina/averages/#temperature
Above are the averages for Taormina. You can use the same website to check any other place in Sicily. However the weather will be similar in all coastal locations at the same time. June is warm to hot. If you really like it hot, then July and August are hotter, but also more crowded and expensive as that is the busiest season.
Before we recommend anything, let us know if you plan to rent a car or use buses, and if you plan to visit around the island or just stay at the beach the whole time. The best beach (voted the best in Italy every year) is probably San Vito Lo Capo. However the Aeolian islands are really great.
Crime is not an issue anywhere. Cosa Nostra doesn't bother tourists, and petty crime is not any worse than elsewhere. Violent crime is actually very low in Sicily. There are actually more murders in San Francisco than the whole island of Sicily (and Sicily has 6 times the population of San Francisco).

I was in Sicily last year June 16 - July 4. I loved it. I live in your area and can tell you that Sicily is hot, though not humid like DC. I'd push for early June rather than later.

I don't care for beaches, so you'll need help from others on that score, but perhaps the following information will be helpful. Check online or in a guidebook to assure that the place(s) you choose have enough historical sights for you. I like seeing historic architecture as I wander, and that is everywhere in Sicily.

Palermo (the major airport, but there's one in Catania, too) is a large, bustling, crowded, noisy city. Not remotely relaxing. It has many incredibly beautiful (mostly Baroque) churches and other religious buildings. But read that first sentence again. If you don't care about the churches and oratorios, Palermo is probably not for you. If you want to see the one best spot, that would be Monreale.

Taormina (much closer to Catania than Palermo) is very beautiful and has beaches nearby. But the town is on the side of a steep hill, and if mobility is a problem, it's probably not a place for you to spend multiple days. Beauty + beaches + nearby cruise port + proximity to Mt. Etna = extraordinarily packed with tourists. The crowds reminded me of Venice and would have driven me nuts on a multi-day visit. If you like the idea of this east-coast location, perhaps a beach-area hotel would work, with one or two trips up to the town either early or late, when I'm sure it is far less jammed with tourists and much more pleasant.

Possible significant: Because Taormina is so touristy, there are day-trip options to other parts of Sicily, including Agrigento (Greek temples, which I did not visit) and the incredible mosaics in the Villa Romana del Casale. The latter is inland and challenging to visit by public transportation (last leg probably requiring taxi).

Do try to get into the interior at some point; don't just stay on the coast.

I liked Trapani, on the west coast and not too far from Palermo. Manageable size and basically flat, with pretty architecture. Buses to a nearby beach, which I didn't see. Big advantage: You can take a funicular from Trapani up to the very evocative hill town of Erice. It's definitely touristy (not crowded like Taormina), but I loved it. Not large but I wished I had a full day to wander around Erice. It is a bit of an uphill walk from the funicular into the town; I don't remember whether there's any form of wheeled transportation available.

There are also ferries to the Egadi Islands from Trapani, running frequently enough to make visiting the islands easy. I went to a couple in one day and just walked around. Nice to visit, but I personally (remember, not a beach person) would have been bored if I had spent the night.

East of Palermo is the pretty little town of Cefalu. I believe RS said it was his favorite spot in Sicily. Easy day trip from Palermo. It's a bit of a walk, I think slightly uphill, from the train station to the town, and the town is on a slight incline. Probably not enough to be bothersome. There is a beach nearby. Although I preferred Trapani (it felt more like a real city to me), I'm sure many folks would vote for Cefalu. Cefalu has a major advantage: Fast ferries running to the Lipari/Aeolian Islands, no doubt more interesting than the Liparis. (Milazzo near Messina is a bigger port for the Liparis.) But check to be sure the Cefalu ferries run in June.

If I were going to stay in just one place in Sicily, I think it would be Ortygia, the historic part of Siracusa, in southeastern Sicily. Ortygia is very pretty, with many restored buildings. Siracusa has an important archaeological part with a Greek theatre and an archaeological museum. There are beaches somewhere (sorry!), and it's a good hub for bus trips to interesting towns like Ragusa (split by a ravine), Noto (famous Baroque architecture) and Modica.

Some corrections to acraven's post, which is very good overall:

Taormina is beautiful but the beaches are pebbly in most places. There is a bus up to the main town. And also an elevator, but there is still a bit of a walk to the town or the beach.

Lipari is the largest of the Aeolian Islands, they are not two separate groups. Lipari has a nice sandy beach area, as does Cefalu. Lipari also has a nice old town but it is uphill from the beaches. Cefalu is an hour by train from Palermo, another hour by bus to Monreale.

You can get to all of the places mentioned by bus and/or train. Villa Romana del Casale can be reached by bus, there are a few short flights of stairs or ramps inside the exhibits.

Ortygia is beautiful but the beach area is small. There is a daily market, and buses that go to the newer part of the island, including the train/bus station.

Agrigento is easier to reach from Palermo, and there is a nice sandy beach nearby.

Erice can also be reached by bus from Trapani.

Thanks, Zoe. I'm utterly useless when folks want to talk about beaches.

Thanks all. Helpful info!!

Here is a great website for info all about Sicily. Hundreds of links to all kinds of info.

Best of Sicily