I think you may be confusing Roman and Greek cultures in this question. While the Romans encouraged athletic events at festivals and were fond of sports like chariot racing, the average citizen participated in these sports primarily as a spectator and the athletes were generally professionals and often from other parts of the empire where athletics had a more established and supported culture (like Greece).

The closest thing to the gymnasiums of the Greeks in Roman culture was the Iuvenes. Iuvenes were local associations of boys and youths drawn from the chief families of their respective townships. Similar to the Greek epheboi, the training given was chiefly military and athletic and designed to train the boys for positions in the Army or Civil Service. Training in the use of weapons and horsemanship were the primary skills instructed, and little is known of other athletic training they did. However, it is speculated that it was similar to that in Greek gymnasiums as many of these Iuvenes used old Greek gymnasiums and palaestra for their headquarters. These Iuvenes represent the only organized attempt in Italy at making physical training a part of education.

In Greece, the culture of gymnasiums and palaestra was an public institutions that served an integral part of the education for all boys growing up in the city-states. The palaestra was technically the wrestling school and resembled an ordinary Greek house with an open courtyard in the center and rooms for undressing and washing. The gymnasium was more appropriately an athletic ground with a running track as the primary focus, but because rooms for undressing and washing were also required they often included or were attached to the palaestra. As these buildings became more elaborate, baths and other amenities were also included.

The palaestra and gymnasium of the Greeks was not just an athletic establishment and more a meeting place for the men of the city. Here, boys are as likely to receive instruction in math or philosophy as they are to receive instruction in physical activity. Some men and boys would likely be playing wagering games or fighting quails before or after their excercise as well. The physical instruction would be at the hands of either a Paidotribes or a Gymnastes. The paidiotribes was similar to a schoolmaster, often had his own palaestra and was responsible for the epheboi or young boys. There would often be assistants who specialized in the instruction of specific exercises. The Gymnastes was often a retired boxer or wrestler who used practical instruction as well as what constituted a science of training regarding diet, massage and exercise to produce the physical condition required for athletic success.

What a typical routine would be on a trip to the gymnasium would be highly dependent on age and station. A young boy up to his enrolling in the Epheboi would likely have been exposed in a progressive fashion at the gymnasium to foot races, wrestling, jumping with weights, boxing, javelin, and discus. At home, he would likely have received instruction in balance, posture and walking. Once a member of the Epheboi, the instruction would become much more focused and directed with wrestling as the primary focus and also included riding and the use of weapons. Instruction in all these activities was done as a series of progressive drills under the direction of the instructor or possibly through musical cues. Dance was often an integral part of these instructions and there is evidence of dances with wrestling or militaristic movements incorporated.

In addition to these more formalized methods of training for boys, any man could come to the gymnasium to practice. Options included wrestling and boxing, running and jumping, throwing, lifting of and jumping with dumbbell-like stones, and ball games. There is very little information on these ball games, but evidence suggests that they were prevalent. Also, swimming and rowing would also be activities that male citizens would participate in.

Everything discussed above has to do with how an average male citizen would utilize the gymnasium. There were professional athletes in Greece at this time and their training would differ based on their athletic focus and the philosophy of their Gymnastes.

As far as diet, only professional athletes would have special consideration for their diet as far as performance was concerned, and there is widely varying information about what this might entail.

Lastly, female participation in athletics varied among the city-states, but there is evidence that women participated in some athletics, primarily in foot races. Generally, the gymnasium was a place for male citizens to interact and exercise without women. Likewise, athletic contest were generally restricted to male participants and make spectators, but this also varied with time and place.

This is a very general overview of the culture of gymnasiums in Greece, but it is by no means comprehensive. I'd be more than happy to answer questions, provide more detail or otherwise elaborate as necessary.

Sources: Athletics in the Ancient World by E. Norman Gardinier ISBN: 0-486-42486-3

Sport and Spectacle in the Ancient World by Donald G. Kyle ISBN: 978-0-631-22971-1

Follow-up question, what did a Roman gym look like? What kind of equipment did they have to train with?

[Adding to question] Is there any book or written documentation of these workouts or diet practices?

In Rome? What time period roughly between 600bce - 400ce