There's a great deal we don't know about Roman gladiator training. Most of our knowledge about gladiators come from graffiti found in places like Aphrodisias (in modern-day Turkey) and Pompeii, murals and mosaics like the 4th century Gladiator Mosaic now in the Borghese Collection and the Villa Dar Buc Ammera mosaics, archaeological excavations of ludi (gladiator schools) and gladiator graveyards, and a handful of written accounts by Cassius Dio, Pliny the Younger and others.
However, these sources tend to focus on the gladiators as people (their number of victories, their fighting style, their status as athletes, celebrities, and sex objects, their diets, etc.) or the games as an institution in Roman life. So we still have some unanswered questions when it comes to their training regime.
Interestingly enough, it's unclear whether or not gladiators did weight training. We have murals showing Roman woman using remarkably modern-looking weights while jogging or doing calisthenics. But there's not much to suggest it was done by men, especially soldiers or gladiators - https://i.imgur.com/M7AJAoQ.jpg
In 390 AD Vegetius writes about the "post exercise" and the use of weighted weapons by Roman soldiers and gladiator trainees:
We are informed by the writings of the ancients that, among their other exercises, they had that of the post. They gave their recruits round bucklers woven with willows, twice as heavy as those used on real service, and wooden swords double the weight of the common ones. They exercised them with these at the post both morning and afternoon.
This is an invention of the greatest use, not only to soldiers, but also to gladiators. No man of either profession ever distinguished himself in the circus or field of battle, who was not perfect in this kind of exercise. Every soldier, therefore, fixed a post firmly in the ground, about the height of six feet. Against this, as against a real enemy, the recruit was exercised with the above mentioned arms, as it were with the common shield and sword, sometimes aiming At the head or face, sometimes at the sides, at others endeavoring to strike at the thighs or legs. He was instructed in what manner to advance and retire, and in short how to take every advantage of his adversary; but was thus above all particularly cautioned not to lay himself open to his antagonist while aiming his stroke at him
So, we know gladiator-style training and military-style training overlapped somewhat. With that in mind, we should consider the next section of Vegetius:
They were likewise taught not to cut but to thrust with their swords. For the Romans not only made a jest of those who fought with the edge of that weapon, but always found them an easy conquest. A stroke with the edges, though made with ever so much force, seldom kills, as the vital parts of the body are defended both by the bones and armor. On the contrary, a stab, though it penetrates but two inches, is generally fatal. Besides in the attitude of striking, it is impossible to avoid exposing the right arm and side; but on the other hand, the body is covered while a thrust is given, and the adversary receives the point before he sees the sword. This was the method of fighting principally used by the Romans, and their reason for exercising recruits with arms of such a weight at first was, that when they came to carry the common ones so much lighter, the greater difference might enable them to act with greater security and alacrity in time of action.
Gladiators also had an assigned fighting style, which was given to them shortly after "enlistment." They could be a heavy, sword and shield armed gladiator like a Samnite or a lighter Thracian with a buckled and curved dagger. Presumably they'd have further training in this fighting style and it's unique weapons. Roman gladiator types had a set nemesis they'd often be pitted against. For example, the Retiarius would use his trident and net to try and catch the Murmillo, who had a fish on his helmet. So it's likely practice bouts would have pitted nemeses against each other.
It's also likely that Roman gladiators spent time choreographing fights. The death rate for gladiators is remarkably low, with only about 10-15% of fights ending in death. Gladiators could also be extremely expensive (if bought as slave or recruited in exchange for debt forgiveness/repayment). This suggests there may have been a degree of choreography to some gladiator bouts. If so, gladiators would likely have practiced such things ahead of a competition.
As for diet, researchers have an isotope analysis of the bones in a gladiator graveyard in Ephesus, Turkey. Surprisingly, gladiators ate a largely vegetarian diet, albeit a high-protein, high-energy diet of barley, beans, oatmeal, and dried fruit. Barley was an especially important protein source, so much so that gladiators are somestime called hordearii or “barley men" in Roman texts.
We don't know if this diet was just regional to Ephesus or universal to the Roman Empire, but it's very likely it was common in other areas as well.
As a result, it's likely the gladiators weren't the shredded beafheads you see in shows like Spartacus. What the depictions of gladiators in mosaics and murals show are guys with a physique like a boxer or Olympic wrestler.
The BBC has a good clip that helps illustrate what we think gladiator training may have looked like - https://m.youtube.com/watch?v=JIWy9Wg7RfY
Followup question: I've heard on many occasions in many places that gladiators actually weren't "buff" the way we think of it (though certainly they were athletic and strong). Rather, their diet and training were such that they would actually carry quite a bit of fat, so that they could be cut in the ring and produce impressive bleeding wounds to impress the crowd without endangering their vital organs or muscles. How accurate is this?