Evidence for women participating in sport and games is significant, however, women were expressly barred from participating or viewing the Olympic Games. The Heraean Games were a single sex series of events in which women competed against each other, divided by age. Greek society as a rule was repressive towards women however they were allowed to participate in sport while they were young. There was a significant difference drawn between a girl and a woman with girls allowed to train and compete while women were, for the most part, barred. The exception to this rule (as it usually is with Greek society) was Sparta. Sparta allowed women to train as the men did while other cities had a more restrictive repertoire.

Training for the Olympic Games feel in line with the training undertaken by citizen soldiers for each respective city-state. The Agoge is often referenced as the training regimen of Spartan society, however, the word itself means training; each city had varying Agoge to train their citizens. Varying cities and their colonies could develop reputations for particular events and as such their training would reflect that.

There were two forms of wrestling during the Ancient Olympics. Palé, standard wrestling similar to today (however far fewer rules and more focus on forcing submission through painful maneuvers) with points won through forcing submission, touching your opponents back if shoulder to the ground, or forcing your opponent from the arena. The Pankration (pronounced pan-kra-tee-on) was the more brutal cousin of Palé with a similarity to modern MMA fighting, with almost no rules. The Pankration had only rules against eye gouging and biting and, with the gradual introduction of the Cestus, became one of the most brutal of events. The Cestus started as a rudimentary boxing glove of leather, however, it developed into a brutal pugilistic weapon with metal/leather lumps or plates woven into the contact surface of the glove (the Romans were involved in developing the brutal nature of the weapon and combat). By this point Cestus fighting was reserved only for slave combatants, often to the death, due to the excessive damage and vicious damage caused by the weapon.

As for women in Roman games, the Gladiatrix became an extremely popular, albeit somewhat uncommon, combatant. Fighting in the arena was seen as the sole occupation for those of the lowest of classes, however, gladiatricies were often from the highest of classes, equestrian and senatorial, which caused a schism of opinion on their use. Septimius Severus even outlawed their use in the Flavian Amphitheatre due to such disputes, although it has been asserted he did so to avoid comparisons between the Gladiatrix and the women of the Imperial Family. This also appeased the conservative element of Roman society and reaffirmed Septimius' inset it's as a defender of traditional values, harkening back to the golden age of Augustine morality and values. Nevertheless gladiatricies continued to be used throughout the empire in arenas outside of Rome.

Training and practices of both the Greeks and Romans focused mainly on forcing submissions through exerting pain and pressure on the opponent. Most of the training was for military service which required high levels of stamina and pain tolerance which directed the training.

I am not as well versed on the particular training practices of the Greeks so I cannot provide much in the way of specifics, however, I will do some research and update what and when I can.

Please excuse any errors as this was typed on mobile. I will also add more sources when I am able to properly quote and cite them.

Sources:

Cruelty and Civilization: The Roman Games by Roland Auguet

Historia Romana by Cassius Dio

The Ancient Olympics by Nigel Spivey

Martial's Epigrams

Thank you. I will definitely find a book or so on Sparta now. So interesting.