Edit: some of these points were already made by Keeler while I was composing this
1.) I only play CSA: I tend toward the smaller brigades and recruiting Artillery separately. I don't have a lot of money in the opening, and I need to spread recruits around the map. Early in the game the 6lbers in large brigades are better than nothing, but eventually end up taking frontage from 20s and Columbiads. Larger brigades take longer to build, (because of the artillery) which sometimes matters. Smaller brigades make it easier to structure divisions the way I want them. OTOH large brigades (the ones in the 1.02 VA force pool have 9 elements!) cost fewer command points than the equivalent number of "loose" elements when they are not in divisions. I purchase large brigades to build new divisions around. I purchase smaller brigades and artillery in order to fill out existing divisions.
2.) I like 17 element divisions in order to minimize command penalties. Makes things a little less flexible but really helps lot in early combat: command penalties really hurt. Also, if two lighter divisions are in separate stacks they may not join each other in combat, whereas putting everyone into one div guarantees that they will participate (within frontage limitations of course). If I had more leaders smaller divs might make sense more often. I still use smaller divs when necessary but fuller is better IMO. Inside a corps or army, I am indifferent to the size of divisions since there are usually plenty of CPs to go around.
3.) As CSA, obviously I like to get as much of the VA force pool onto the map as quickly as possible, usually at Alexandria (if I have it) or in the Valley at whichever city is safer. If I am not doing well, everything gets built at Richmond. Either way, VA doesn't have 10lbers, so those get built in NC. I build (and organize) at Memphis (TN pool) and Corinth/Holly Springs (MS pool) for defense of the river, and at Nashville (TN pool) to prepare to rush Louisville. As soon as I get KY, I concentrate on building in Bowling Green, or better yet, Louisville because I may not have access to the KY force pool later in the game. I tend to go straight for Louisville to get a forward build point right in the heart of the Indianapolis/Columbus/Lexington/Cincinnati cluster (all strategic or objective cities); troops built in KY are able to join this fight quickly, and the KY force pool is very large. It is critical for the South to get and hold Springfield MO early in order to build (the limited) force pool in MO; otherwise everything has to trickle in from Memphis/Little Rock by rail, river and a walk across the Ozarks. Most of my Gulf reaction force gets built in New Orleans, training up under Bragg while they wait to see where the Union will land (I always build a couple of sea transports there so I can move the QRF freely along the Gulf Coast.) In TX I build wherever is needed, but I hardly build in AL/GA/FL/SC except to garrison or possibly as a QRF for the Atlantic coast (my lowest priority against Athena).
I usually form up planned divisions as soon as the leader I want is active, even if I am only putting one element in it to start: more troops are coming, and when they get there I don't want to have to wait around for activations to get organized. I try to keep my best 1*'s in charge of divs and so send them to where they will need to be ahead of time, even if the troops aren't ordered yet. If I am playing with hard Activation Rule I try to form divisions within armies or corps if I can: they are prone to going inactive and becoming fixed for several turn when formed on their own. As the CSA, training happens as close to the front as possible. I only have one leader that can upgrade conscripts so he goes to Louisville or wherever I am doing the most building; he can only do two elements per turn, so I keep him at my busiest forward build point to upgrade the conscripts in new brigades for immediate use. Occasionally I will have him train some militia, but this is more so that I don't have to commit whole brigades to control captured cities than for the $ savings. HQs and Training Masters (the experience guys) go to large front line stacks to maximize the number of elements they affect per turn, except for Bragg who trains everyone in place at New Orleans (he isn't much use for anything else).
I hardly ever purchase cav outright for use in divisions, there are plenty of brigades with cav attached. Individual cav elements get built for scouting/MC busting stacks (2 cav, 1 cav art), and I never have as many as I need/want either! I have not had the combat success with large cav divisions that I had in AACW, so don't use them much anymore except in the case of the Laurel Brigade (it costs 4 CP anyway, so it might as well form a division with a couple of extra cav artillery).