Catania is a city on Sicily.
South towards Syracusa
First Option: Take a bus out towards the airport. Get off the bus as soon as you can after it's come off the big spaghetti junction area. Walk back up and over the junction on the lane with the traffic going away from the airport. Take the first slip road on the left for the SP70, (signpost - Siracusa/Ragusa). Follow it under the bridge until it meets with another slip road. From there you can hitch both slip roads on the white line area.
Second Option: you can also walk 1.5km south of the city, to the roundabout at the end of Via Plaia, at Via Domenico Templo. There you can go to Siracusa by motorway (just stand on the sliproad, there is a little bit of space, it's called Via San Giuseppe alla Rena), but traffic goes via the tangenziale, and therefore in all directions. Eventually, however, you should find somebody, even though you might be offered rides to North and East directions, too. The second possibility at this spot is taking the Statale 115, but it goes via all the beach resorts which might be a bit cumbersome. We tried both options and got a ride after 30-45 of walking back and forth.
Take a bus out towards the airport. Get off the bus as soon as you can after it's come off the big spaghetti junction area. Walk back up and over the junction on the lane with the traffic going away from the airport. Keep to the right and follow signs to Palermo to meet the top of the A19. It's a bit of a walk to get to the slip road, maybe twenty or thirty minutes. Alternatively you might be better off getting a bus to Librino and walking to the slip roads from wherever it stops, but I'm afraid I don't know the bus routes.
North towards Messina
From the centre, walk 2.5km north to the Motorway exit Catania Est which is in the North of the City (there might also be a bus, we haven't checked). You can comfortably stand on the Via Vicenzo Guiffrida with a sign, some 100-200 m before the actual motorway starts. Cars going to Messina have to stay on the left lane, but in Italy you'll always find somebody willing to pull over at the risk of causing minor disruptions, so it's still a fairly good spot.