Example: During Octavian's rise to power, he had to raise several new legions to fight Caesar's assassins. From what I've read, the troops included a lot of veterans, so training wasn't a huge issue. And the money was provided by Caesar's will.
But where do you just acquire thousands of standard-issue swords, shields, pila, etc. that a first-century BCE Roman legion would require, on such short notice? Were there stockpiles just sitting around? Or would he have to hire every swordsmith and armorer in Rome? A previous question theorized that veterans probably didn't keep their equipment, and if so Octavian would've needed to resupply them all himself somehow.
(This is specifically about post-Marian Rome, where the state provides the equipment, rather than the other method of "if you want a shield you better buy the damn thing yourself")
This is a pretty excellent question, and I hope I can clear things up for you! Before I start talking specifically about Octavian/Augustus' levies, I'll go ahead and talk about the most heavily neglected aspect of any military discussion: logistics. That boring thing that makes sure that your men are actually able to fight when you need them to? Yeah, that'll factor heavily into play here.
The Romans were - and still are - recognized as master logisticians (even with people like Julius Caesar, who was a tiny bit useless on that front), and often won their victories or took their territory out of the sheer power of being at a place, in force, well-equipped, well-fed, and well-rested and organized, even while their potential opponents were still working on step one. They took copious notes and made receipts for everything, some of which can be rather interesting (like the number of sick men in an army on any given day). Unfortunately? We've lost most of them because people think that logistics are boring and that receipts should be trashed. What we do have, though, are a good number of records detailing one specific aspect. Unfortunately, it's not arms and armour - it's more important: food. With the records about grain donatives, we can kind of piece together a picture of how the Roman system worked, especially during the First Century BC, when everything was a delightful mess of fucked-up-ery.
So how in the name of Mercury did a general supply 200,000 men whom were raised within a month or two? Well, there were a few ways. First, and least complex, the general in question would buy local food. As the least controversial method, this one would keep the locals kind of happy, even if he would only purchase things at a set cost. Locals would agree to that, because getting money for their contributions would be an excellent alternative to the soldiers murdering all of them and taking the food for themselves - along with any local valuables, women, etc. This was also one reason that aspiring warlords quickly made moves to consolidate their respective war chests as quickly as possible: money is the fastest way to any man's heart. Some allusions to this practice are from Sallust, when he describes Pompey buying supplies on credit, and in the invective against Sallust, where the authour (Cicero? We dunno) accuses Sallust of doing the same thing for Caesar.
Other options were donations - some allied cities and provinces donated generous amounts of food in an attempt to curry goodwill. Unfortunately, this almost always backfired, since the Romans expected these "donations" to continue coming, as was the case with both Carthage and Syracuse. More profit-minded individuals would attach themselves as sutlers - merchants who tagged along with the army, selling them food and other supplies at a healthily marked up price. Finally, the regular methods of Roman taxation could squeeze the provinces of which the general had control (Men in charge of Roman armies were always politicians, and were assigned to rule provinces at the start of every year - at least until the system broke down).
So what does any of this have to do with arms and armour, you might ask? Just replace the word "food" with "weapons and armour." Oftentimes, jobs like these would be given to contractors, with an arrangement of "I'll pay X amount of money for Y swords." How the contractors got the job done wasn't necessarily a concern of the man in charge - the concern was that his men got the weapons they required. The most important note here is that the Romans...well...didn't raise armies out of thin air, however much it might seem that way. The Roman campaign year began in March. Here's how Polybius (2.23.9-11) describes the preparations:
[The consuls] enroll[ed] their legions and ordered those of their allies to be in readiness. . . . Of grain, missiles and other war materiel, they laid in such a supply as no one could remember had been collected on any previous occasion.
^^Translation ^^borrowed ^^from ^^Roth's ^^The ^^Logistics ^^of ^^the ^^Roman ^^Army ^^at ^^War. ^^It's ^^200 ^^years ^^before ^^the ^^time ^^we're ^^talking ^^about, ^^but ^^it's ^^an ^^excellent ^^example ^^of ^^the ^^planning ^^that ^^goes ^^into ^^these ^^things.
Preparations had to happen before armies were raised. Supply depots had to be created and filled. Money, food, and arms were gathered and dispersed accordingly. A bureaucracy had to be formed and established. Recruitment parties had to be formed. You'll note that one of the few times that a Roman brags about being able to raise an army from nothing ("All I have to do is stamp my foot and thousands of Romans will rally to my call"), he fails miserably and is forced to flee the Italian peninsula to properly raise an army (See: Pompey). Finally, by the time of Augustus, the Romans had not only been involved in constant foreign wars throughout the past 500 years, but also constant civil wars for the past 70ish. The weapons and armour for his armies didn't pop out of nowhere - they were already there, and just had to be purchased or requisitioned.
For further reading on Roman logistics, I highly recommend the aforementioned Logistics of the Roman Army at War. It's a comprehensive study of Roman policies in the matter through the Republic and into the middle of the Imperial period.