With the exception of the origional Olympics games from 776 BC, there seem to be few things we would call 'sports' now from the Roman or Greek classical era.
The pentathlon was made up of running, long jump (no run up), discus, javelin and wrestling. The foot races were over an agreed distance (I think the equivilent of about 200 yards, but I'm doing this from memory). There were also chariot races followed by horse racing, with the next couple of days holding events like boxing (no gloves), more wrestling, and more foot races.
There were other games which coincided with the origional olympics - the Pythian Games at Delphi, the Nemean Games at Nemea and the Isthmian Games at Corinth. All were held 4 years apart, and on similar lines.
Other than that, most if not all physical activity seemed to have some military root. When armies begin to be 'full time' rather than seasonal a few other things come into play. Sparta for instance, which had the Helots to do all the farm work, had rigorous physical criteria for men to become full members of society, and they worked at it all the time, and as long as they lived.
Famously, the Roman army from the time of Marius was said to have conducted 'drills like bloody battles' and 'battles like bloody drills', such was the constant training and quest for discipline. This was necessary and not surprising, as serving soldiers on campaign might need to carry 90lbs all day, every day.
There was only one sport as we would think of it in ancient Rome - chariot racing. This had a huge industry surrounding it, everything from trainers and vets to gambling rings and prostitution. Members of Roman society were fanatical about which colour they followed (red, blue, green, white origionally, but later others were added). Riots following unpopular results were fairly common, which I think was one of the reasons the first Fire Brigade in Rome was set up.
