Roman cities contained baths that more than 'often' acted as sports facilities, they were whole athletic centres. These buildings were often a centrepiece of municipal life. They would contain spas, wading pools, and an atrium for general workouts. However, many also included ball courts and areas for martial activities, while outside the baths, the Campus Martius was an area widely used for foot races. In general, the physical activity that Romans found enjoyable would have been popular among pleb and patrician alike. It is also important to keep in mind that for the upper class, leading soldiers was considered one of the penultimate goals and thus keeping yourself in shape through martial training would have been a large occupier of time for many in the upper class. We can assume that a lot of the core muscle strength was built up through those types of games, like boxing and wrestling.

In terms of diet knowledge, the Romans were not ignorant here. The Gladiator schools were entire facilities designed to create top physical specimens, and this is knowledge that a Roman aristocrat would have had ready access to. The work-out regimens from such a facility would also likely have been beneficial to anyone interested in learning.

All that said, there are two caveats to this. First, very little archaeological or literary evidence survives attesting to the rigour in which Roman aristocrats applied themselves to the goal of getting into and staying in shape. It's clear they would have had the means based on our evidence that the Romans knew how to through their military and gladiatorial experience. Whether this was applied in a consistent manner in order to achieve desired goals that might be common in today's gyms is not clear based on the evidence available. At best we can use our keen powers of observation to extrapolate that, at least among younger men, it was very likely staying in physical shape would have been high on their priority list. This also changes depending on the era as well. During the late Imperial period, Roman aristocrats were not the martial creatures they once were, while during the Mid-Republic during the height of their expansion, it is very likely the martial arts and physical fitness were a much higher priority, particularly given the reserved (almost Laconic) nature of Romans during the early and mid Republic.