The death of Socrates might give you the impression that classical executions consisted of downing a cup of hemlock, and then drifting off painlessly during an erudite discussion on the finer points of philosophy. Nothing could be further from the truth. Hemlock was the execution method for the upper class. Common criminals could expect somewhat harsher treatment.
The laws of Draco specified death for virtually every crime from murder, to shortchanging a customer in the agora. By classical times the Draconian laws had been repealed, but from the number of times they get referenced in court cases it's a fair guess people still had regard to them.
If you were an average convicted crim, you could forget about hemlock. More likely you faced a complex stock which trapped your wrists and ankles. A metal collar was then placed around your neck, and this was progressively tightened until you strangled, or your vertebrae snapped. This is very similar to an execution method popular in Spain up to Napoleonic times. The Spanish too used a metal collar that used a screw to tighten it to either strangle or break the neck, depending on how quickly the executioner worked.
I've read a mild suggestion that there was an execution ground outside the city, on the right hand side of the northern road to Piraeus. I think it certain that such executions must have been carried out outside city bounds, but I know of no archaeological evidence for a location.
It's possible there were stonings for some particularly heinous crimes. If so, then killing your father would have been one of them. There are known rituals in which stonings were acted out. Certainly other cities had stoned their citizens.
The Greeks were much more spear people than sword people. But spears make rotten weapons for executions. There are documented military executions done using swords. Since Greek swords were rather short, this would have been up close and personal. Probably the victim kneeled and was either struck in the back of the neck, as per a mediaeval execution, or else the sword was thrust from above into the heart. Think something like the early scene in Gladiator where Maximus is taken into the woods to be executed (and promptly kills his captors).
The Athenians had one very interesting attitude totally different from other civilizations: they were just as happy if a condemned prisoner simply went away. If someone wanted to run, thus exiling himself forever rather than face death, then that was just fine. In fact they seem to have almost gone out of their way to leave the prison doors ajar. Clearly the Athenians were not a particularly vindictive people, and equally clearly they considered that to no longer be a citizen of Athens was a fate every bit as bad as death, which tells us a lot about how much they valued their citizenship. The number of men who chose to remain and die rather than lose their city is quite amazing.