Your suppositions are mostly correct.

One important one isn't, though: once we move beyond the early republic, legions were not raised and disbanded annually. Rather, legions were raised for a specific campaign and then generally kept in service until that campaign was finished. The legions would be renumbered at the start of each year, but they were not disbanded and reformed. They might even receive reinforcements after a few years to bring them back up to strength, though it was also common for severely under-strength legions to remain in the field. The legion would then be disbanded at the close of the war, or at least when their part in it was finished.

To name one extreme example: the legions formed out of the survivors of Cannae continued to serve, in some cases, up to 20 years after that event. This was exceptional, but service for several years at the time was common.

Now, as for the process of recruiting and training:

Yes, many of the men in the legion would have served previously. Indeed, in the mid republic, legions were organised by age-group, and those serving in the second and especially third line of the heavy infantry (principes and triarii) would likely have served in quite a few campaigns before. (Although that did not necessarily mean they'd been in any big battles before. Battles were relatively common in Republican times, but in some theatres campaigns tended to consist of raiding and sieges instead) However, even such men with previous experience would not have served together as a unit before, and still would require drill and training before they could effectively operate as an army.

Yes, legions were typically trained and drilled before actually moving on the enemy. This was one of a general's primary tasks, and the good general training and disciplining the men is a very frequent topos in the literature of the period. (and afterwards, for that matter.) We do not know how exactly this took place in practice, though: there are no technical military manuals or any other kind of detailed description of how Roman armies were raised, just off-hand remarks made by ancient authors while describing other things.

To illustrate, let's take another example from the second Punic War. When Scipio was authorised to invade Africa, he first went to Sicily to prepare for the invasion. We're not sure exactly how many soldiers he had — See this post by yours truly for an in-depth discussion of the sources for this example — but we do know he brought with him 7000 newly recruited volunteers and added these to the Cannae survivor legions I mentioned above. These volunteers were used to bring the (by now thoroughly depleted) Cannae legions up to full strength and then some. It's also probable that he brought a similarly sized force of Italian allies with him, as Roman armies usually did in this era.

Scipio arrived in Sicily in 205 B.C., but did not invade until the following year. In the meantime, he arranged for a fleet and tried to win allies, but most importantly he needed that time to bring the newly recruited volunteers up to speed and integrate them in the existing veteran formations of the Cannae legions. Even those veterans may have been in need of training: although they had been serving for absolute ages by the standards of the time, the war in Sicily tended to focus on raiding and sieges and they had not taken part in any major pitched battles since the disaster at Cannae a decade earlier. This period of preparation was typical for any major expedition in the era.

One thing to note is that all Roman men eligible for military service were expected to know how to use their weapons already, so it doesn't seem that this would be a big part of the training. They'd practice as boys, and be taught by their fathers. It's only later when the Romans start to recruit men from previously ineligible classes of society that Publius Rutilius Rufus hires gladatorial doctores to institute a system of weapons training for the men.

Discipline doesn't seem to have been a problem. The Roman legions were notorious for their strict discipline and draconian punishments even in Republican times. Citizens who enlisted gave up many of their legal rights and protections and could be flogged or even executed without a trial. (Execution-worthy crimes included cowardice and sleeping on guard duty.) The Romans' willingness to accept this goes a long way towards explaining the success of their army despite its semi-professional status. (Most Greek hoplites were notoriously unwilling to accept such discipline, for example.)

Despite all this, it's quite clear that newly raised Roman legions of the mid-Republic weren't nearly as effective as forces that had been fighting together for some time, and that disbanding a veteran legion at the end of a campaign meant that much of this fighting capability would be lost. There are quite a few examples of newly raised and insufficiently drilled legions meeting with disaster, with Cannae itself being the most dramatic example. This was an inherent trade-off of the Roman militia system.

Sources:

Adrian Goldsworthy, The Complete Roman Army

Adrian Goldsworthy, The Fall of Carthage

Paul Erdkamp, A Companion to the Roman Army.