Where to begin.
The CS army will not be overly big. For a start, they have a much smaller manpower pool than the US has, made even smaller by the fact they refuse to let a very large minority have equal rights to the majority. A CS army will be a relatively small federal thing, at an exceptionally rough estimate I'd say no bigger than 20,000 men maybe even smaller. The states themselves would be defended by state run militias (National Guards if someone gave them more power), and each guard will be loyal to their own state rather than to the Confederacy as a whole. So a war that will only immediately benefit a few states will not have the support of the others.
A CS navy would be a similar dealie to the army, a small federal institute supported (in theory) by the state run counterparts.
The CSA would have an exceptionally backwards economy, it being mainly rural and all, and even if Britain and France decided that the Confederates were not only their bestfriends but were worth wasting money on supporting, the South would still never be able to bring an army up to scrath with the USA, or even Spain for that matter. And no, the Royal navy will not be sailing to the Bay of Pigs anytime soon.
The CSA has to have a larger army than the Union does, to continue slave patrols. If it comes to keeping their money intact the planters are just peachy with paying for the army required to sustain that. This means that after the 1860s the CS Army pretty much starts straddling the line between the Republican Guard and a conventional military, and arguably does not see necessarily employment or training in more modern, conventional combat roles even without a CS military putsch.
The US army was pretty tiny, I think the first time it had ever made it into the millions was for the Great War. Before that it was a standing force of some 16,000 professionals who rode around the west beating up Indians.
Both armies are naturally going to be larger, but as armies are seen as tools of big, oppressive, centralised governments, and both countries constitutionally have to avoid those, they're still gonna be fairly small.
If anyone can find figures for the US standing army around 1880-90 I'd actually be much obliged.
The US military system of the time was rather wretchedly complex, so it's difficult to provide "actual" numbers. For one thing the US Army of the time had what it called "Volunteer" regiments which were precisely that: mass armies of untrained volunteers. These people were people without any real exposure to Army life whatsoever, and this created major issues of needless casualties in an unvarying, monotonous, and depressing fashion for the United States.
The Union Army of the US Civil War was 1.5 million strong, but in terms of the structure, it was created by convoluted patterns of enlistment, as well as recruiting from both US states and states controlled by the so-called Richmond government, such that the Civil War is one case where numbers can only be estimates. Thus while I say the Union army was 1.5 million strong, it might have been larger at various points, or alternately it might include in several cases counting one person twice or thrice due to multiple enlistments.
Further complicating this was the pattern of Brevet ranking, where people got all the responsibility but none of the extra pay, and which could and did create issues where people who were Brevet Major Generals of Volunteers would be outranked by Brigadier Generals of Regulars, and who controlled what when was a massively convoluted prima donna game.
Thus while the US *Regular* army was tiny, the total mass of the US Army was not entirely so much. The system to me resembles in intention what the Soviets tried to do with their concept of a mass reserve, and like the USA the USSR never really made *that* work either.
The US Army of the antebellum and to some extent the postbellum era was not only tiny, but it was distributed at multiple forts, it was required as well to garrison frontiers next to remaining Indian territory (not by any means easy duty), and it was underpaid and led by old fossils due to absence of a pension plan and the Army providing perhaps the only profession at the time where people could be cared for until they croaked.
It was a very different world for the military then, and the prospect of the 1850s, this time with an actual army behind the slavecatchers is for the USA what Germany defeating France was for the OTL USA in the 1930s: those days are over, a much larger army is *required* to sustain itself. This probably leads to a lot more 1877s and Bonus Army Incidents in the USA, too, as there's more army to throw at those problems and much less fear of having a big peacetime army.