Warning, long post.
The are three things you have to take care of on the early turns as the CSA that are not immediately apparent: Ft. Sumter, Norfolk and Rolla (Sumter and Norfolk have been mentioned by others). Sumter is obvious, just use the infantry under PGT Beauregard; if you don't do it right away it becomes much more difficult. Second, on the first turn that Virginia secedes, take Norfolk using the Suffolk Militia that is stationed one region west of Norfolk; if you do it immediately (be sure to use the Red, Assault, posture so that it takes the town) the Union is scripted to abandon it without a fight. Third, on the first turn that Missouri opens up, build one militia unit in Rolla, then on the next turn (milita are built quickly enough they can begin to act on the next turn) destroy the depot there; you won't be able to stop the city from falling into Union hands a turn or two after that, but without the depot they will have a hard time supplying operations against Springfield.
These three tasks are important no matter whether you play against a human or the AI,. and the ability to do them pretty much goes away if you don't do them immediately. The advice given by others of playing a couple test drives and of making a turn checklist is excellent. Once you are playing for real, here is what I do against the AI. (There are many other good strategies for all of these theaters, and you won't be able to accomplish everything below in every game, but if you can get a good portion of this stuff done you should get through the first year of the war in a reasonably playable position.)
Regional tasks for the first few months of the war, West to East:
FW
Build three or four Rangers in Adobe Wells TX whose targets are the two stockades in between Denver and Taos, NM (Ft. Lyons and the other one I think is Ft. Collins?). If you can capture and then destroy these two stockades the Union will have difficulty bringing new artillery into New Mexico, making the defense of El Paso easier and cheaper. Eventually, build a regiment of regulars and a 12 lber in El Paso. These, plus the scripted forces should be enough to defend El Paso against all but the most determined push against it in the early years of the war.
MO/AR/IT
(Make sure to destroy the depot at Rolla.) As quickly as you can, get everything in the area up to defend Springfield: Nathan Lyons is going to attack it in the late spring/ early summer. If you have destroyed the depot at Rolla and can repulse his initial attack then you should be able to hold Springfield for quite a while. If the Union takes Springfield you will not be able to build any troops in Missouri, instead having to build and move them from eastern Arkansas. (Also, MO has artillery in its force pool, and AR does not. You will need them if you are going to contest the rest of Missouri.) Further, Fayetteville is dangerously exposed to a Union-held Springfield. Consider capturing and burning the Union stockades that lead in a line northwest from Fayetteville, as they provide cover and supply for troops built around Leavenworth KS to threaten your western flank around Fayetteville and Springfield. Stand Waite and his Indians are a good for this, as are the Mounted Volunteers (who are not cavalry, but fast infantry).
Gulf Coast
The Union starts with troops and transports at Ft. Pickens (off the coast from Pensacola, FL). Some time in the first few months of the war, the AI will take a stack of 200-300 PWR from there and attack somewhere along the Gulf Coast. They are fairly easy to fend off, but you don't have the resources to fortify all of their potential targets. In order of priority for you to defend are New Orleans, Mobile, Matagorda TX (the Cotton Bale region) and Pensacola, FL. New Orleans will get a scripted brigade of militia eventually, but this may not be large enough or quick enough to stop a quick attack from Pickens. The AI is wary of the forts guarding Mobile, but it is too valuable to rely on these alone. A brigade of regulars in Matagorda is plenty, but you may not be able to afford it. Since you can't stop them in all four places, you are likely going to leave Pensacola the weakest, so expect them to take it or the regions around it. This is not an immediate problem, as that stack cannot do much to hurt you right away, but over the course of the game it will become a big problem: it is close to Mobile, and on a rail line going north to Montgomery and northwestern GA. In Union hands troops and supplies can flood into the theater later in the game under the protection of Ft. Pickens and seriously threaten the entire south east. Unfortunately, you may have to accept this long-term liability to defend the more immediately important regions.
Mid-South (TN, eastern AR, northern MS and AL)
There won't be much going on here right away, because you are somewhat protected as long as KY is neutral. Small Union forces from Cairo will often harass southeastern MO and northeastern AR, but it may not be worth the effort to oppose them. Steadily build forces at Nashville, Memphis and Corinth to thwart any shenanigans in TN. Consider stationing troops and artillery at Fts Donnelson, Pillow and Island 10. Again, you won't have enough to fortify all of them, and Island 10 in particular can be tricky since it is hard to get in and out of (swamp) but all three can at least bombard passing Union river forces and interdict Union supply flow along the rivers.
Sometime in the winter of 61 KY will declare for one side or the other. Louisville is vulnerable to a lightning strike against it from Nashville/ Bowling green if you are prepared. The AI will react poorly to this and will rarely invade TN afterwards since you will be threatening IN and OH from northern KY. (This is very risky against a human, however: you will be dangerously overextended, so this may not be a good habit to get into.) Bowling Green is vital to your efforts no matter what: from it you can access the HUGE KY force pool, and it sits on the rail line that leads from the strong Union build points at Cincinnati and Indianapolis into northern TN.
If you are playing a more defensive game in this region, seizing and then building a fort at Paducah KY will make defending western TN against invasion along the rivers much easier. This is a medium term goal (as is invading KY) but requires preparation during the summer of 61 to accomplish.
East
This is a tricky area, play conservatively. You will always be outnumbered here, and losing a few major battles can quickly cause a downward spiral that you cannot recover from. It is easy to lose the war early through incautious play here. I usually try to take Harper's Ferry with the Army of the Shenanodah. If successful, it will cause the Union to split some of its forces off from Alexandria and DC to try to take it back. With clever play you may be able to take Alexandria behind them from Manassas, allowing you to establish a strong defensive line along the Potomac. Be sure to leave a moderate force inside Manassas so that a Union stack cannot come in behind you and take the depot in a single turn (without capturing the depot they will be cut off from supply and not able to stay for long). Keep an eye on Fredricksburg VA, which is along the direct route from Alexandria to Richmond. Allowing the Union to maintain a foothold there will necessitate pulling almost all your troops south to defend Richmond, leaving northern VA at the Union's mercy.
If you choose to play more defensively, the Union will try to take Manassas early, as well as Harper's Ferry. You may or may not be able to get/keep HF, but you definitely need to hold Manassas. You should be able to mount a strong defense there and hold it without too much trouble.
You will need to begin beefing up your forces in VA right away (yes, there are too many things to do and not enough resources to do them, but no matter your strategy on the rest of the map, not losing in VA is the top priority.) You can build really big brigades from the VA force pool that perform really well in combat, but they are super expensive and take a while to build; you may be better off with several smaller ones, it is one of the things you will get a feel for with experience. You will need lots of artillery in this theater, but you will have to build most of them in North Carolina, since the VA force pool is limited in artillery. Since you will primarily be defending, the 12 lbers are economical and good for defense in the beginning, but eventually they will be outclassed and aren't ever very effective in offensive posture.
Keep an eye on Ft. Monroe; even moderate defenses make the Peninsula a tough row to hoe for the Union, but if you do not take precautions you are leaving the back door to the capital wide open. If you put a couple of brigades (and some artillery to bombard the estuary) as a token force in Norfolk the AI will usually leave it alone. A human player on the other hand will probably not leave such a juicy target unmolested, so be prepared....
Establishing the Potomac Line like I do is probably an intermediate strategy, you might want to instead turtle around Manassas and the in the Valley your first few playthoughs until you get a feel for things. Another excellent intermediate-difficulty approach is the All-East strategy promulgated by Gray Fox. Do a search on the forum to find out more; he and others explain it better than I can, and it is an important concept that you should both be aware of and try out at some point.
I left out some stuff, West Virginia in particular, but this is probably already too much for one post as it is. These are in no way shape or form the best strategies out there, nor will this stuff probably give you good results against a human, but hopefully it will help in figuring out a workable plan to get started with

.