You can take a boat that in 40 minutes will bring you to a sand shore, where there is nothing but the beach, beach equipment, cafes, bars, restaurants and discos:) The place is called Uchkoo'evka:) funny name though:) The boat goes every hour, and the last boat from Uchkoo'evka goes back to Sevastopol' at 7 or 8p.m, so don't miss the last boat if you don't want to stay there for the night;)
From Sevastopol' on board the ship you can get to Istanbul. :)
I took a bus, a trolley-bus, and 'routed' taxi to get around. I did not see trams. You can buy a ticket right in the bus - a conductor will ask you to do it. It is very cheap, and it can also be a pleasant experience, because often the conductor is a pretty young woman. I would also recommend you to go walking when you get to the center, it is not overcrowded by cars.
I do recommend to use ships, skiffs, cutters, boats of all kinds to get to any Crimean port, especially on the South coast, because the scenery is fantastic.