I did this hike as a part of my Sicily trip during June I would suggest to go into the spring/summer and connect it to your vacation on the island. 

There are many organized tours one can book in advance - from really going up the south active crater to sunset hikes etc. The one to Torre del Filosofo can be done without a guide. 

Things to know:

Weather on Mt Etna changes a lot and it is best to have sturdy shoes like trail running or hiking boots to enjoy the volcanic scenery the best. Cap, sunglasses, sunscreen and it is windy so: make sure you have proper clothes with you i.e. windjacket, puffer. :) 

If not coming by car, there are organized bus tours from individual cities, but if you like to be more of a solo orga hiker like myself -  there is a daily bus going from Catania to Mt Etna an back. Downside is that is is only 1 a day so make sure to get on it on time. 

Directions via public transport 

Best way from Catania is to get n AST bus from central bus station (next to train station), bus stop n.6 for a return ticket of 6.60€. Departure at 8.15 am but be sure to be around 7.30 am as the bus leaves with about 40 people no more. It is only this one bus daily.  

Buy the ticket over AST ticketing app the night before so you have no issues inding the ticket shop or buying it on the bus.

Ride has a short stop at Nicolosi for 20min for coffee break & to pick up local tourist guide that will inform people they can get a guide or ride quads for a certain price. 

Arrival at 10.15. Departure back to Catania - bus driver says 4pm but I would advise to be around 3.30pm bellow just in case. The last cable car Down is at 3.50pm.

There are three options you can do the hike - 1) from Riffugio S.1900m to 2900 on foot ( I saw some people doing it but the to be honest the way to Fuvina up is rather dull hikewise with volcanic ash and the time you would use hiking up you can spend up nearby Torre del Filosofo enjoying the beauty of being nearby an active volcano, but be my guest) .

Or 2) get the cable car round-trip ticket to 2500m(52eur) or 3) cable car + 4×4 with buses to 2900 and a guided tour by a volcanist/guide(80€).

The Furnivia Etna website will say that only with buses you can go up but I got local tip from the tourist biro and also saw other locals hiking up. If you dont have the proper stamina and gear then opt for the full service as they will provide you with the gear and it is a different experience than hiking up 500hm. :)

Tip: book your cable car ticket online to save time as it speeds everything up and you do not have to queue. 

Once at Fuvina you have about 2h or so to get to the Philosopher point. That's the furthers you can go without a guide. 

There are options to go further and they are organized mostly for a certain price over 300€ depending who organizes what.

Make sure you check out the situation on Mt Etna as it erupted a week after I was there.

The lava flow is super slow so it is not an immediate danger, however there are other particčws in air and smoke etc that can pose some danger so be sure to inform yourself prior the hike