It depends on what position someone was going to hold within the army but in general being an officer in a Roman army involved a lot of 'on the job learning'. There were a lot of changes both subtle and drastic to the structure of the army from the Polybian legions of the early republic to the Marian reforms of the Late Republic/Principate to the Dominate and the fall of the Western Empire. I'll try to explain a bit about the training of officers during the Principate, from the 1st to 3rd centuries AD.

The senior leadership of the Roman army during the Principate were made up of nobles serving their turns in the Cursus Honorum, the series of steps to increasingly prominent offices undertaken by ambitious Roman nobles. The commander (Legatus) of a legion would be a Senator (the most elite of Roman nobility other than the Imperial family themselves) in his mid-40's who would have briefly served as a lower-ranked officer in his youth and since held a series of governmental positions. His second in command (Senior Tribunus) was a younger Senator (usually a family-member or friend of the family) who would be serving as an apprentice. There would also be 5 Junior Tribunes attached to the legion who would be equites (the second rank of Roman nobility) in their mid-30's who had previously served as officers in the less prestigious auxiliaries.

While the senior command of the legions were amateurs the backbone of the command structure, the centurions, were hardened professionals. The majority of centurions had joined the army at the bottom of the food chain, as legionaries. While they had little professional training as officers they had to pass brutal basic training just to join the army. From there a legionary who proved himself could advance through a series of positions with increased prestige, responsibility and pay. On average a legionary would take 16 years to make centurion. The centurions have gone down in legend as ruthless taskmasters and this was generally rooted in fact. Centurions had great authority in disciplining and rewarding their men, lived with them in the barracks and fought at the forefront of battle to inspire them. From a basic centurion it was possible to be promoted further into the elite first cohort and again to become Primus Pilus. The Primus Pilus was the most highly ranked centurion in the legion and it was a position of massive prestige. From this position the final advancement possible was to technically leave the army and become a Prefect of the camp a member of the noble-dominated senior staff. This position came with a noble title and was one of the few ways for a commoner to join the aristocracy in Roman society. The Prefect of the Camp was the third most highly ranked position in a legion (behind only the Senators) and is often thought to have been in charge of most, if not all of day to day business of a legion and as such it was a position of immense responsibility.

tl:dr while there was no school of excellence for Roman army officers the backbone of the army's command structure was still professional and effective due to long years of service in the ranks.

Source: The Roman Army by Adrian Goldsworthy