Which Army CO branch offers the most opportunity for additional schools, training, and promotion?

In being sure to make 0-4 to O-6 The Logistics branches OD, QM, TC and LG have more mid grade officer opportunities than combat arms.
Personally getting a good combat arms branch as a company grade officer than switching over to logistics gives a broad opportunity and perspective on how the Army works!

Promotion goes to Combat Arms, Infantry and Armor, they are just the biggest. That said, most long term officers end up with an additional branch. As far as school, once you commission you are on a path that has some pretty demanding school. Once you commission you will be sent to your Branch Basic Leadership Course. Infantry officers going to Light, Airborne, or Air Assault units generally get sent to Airborne School, then to either RTAC and then Ranger School, or just Ranger School. Then you get all your self help stuff and broadening assignments, staff duties and platoon leader, until you go for Captain. Then it's MCCC and a chance at Ranger School again if you didn't get it after the Basic Course. Then your next promotion step is the Intermediate level at Command and Staff College, put any assignment related schools like Stryker Leader or Bradley Leader. There is a lot of ASI type courses available too.

EOD by far as the longest BOLC and additional training. As far as promotions, I would think the branches that are growing the most offer the best promotions— FA, ADA, Cyber?

Combat arms by far. More combat arms (Infantry, Armor, Engineer, Air Defense Artillery, Artillery) officers and NCOs get more opportunities for training, locations, and yes, if you're good, promotions, than the rest of the Army. Some Combat Support (Logistics, Signal, Quartermaster, others) get better than average chances but the majority of the "good stuff" goes to Combat Arms officers and Soldiers.

Infantry

Infantry would most likely have the opportunity for schools (Airborne, Ranger, etc.). Promotions go at about the same rate for most officers. Some branches are so small that opportunity for upper level promotions are limited (Finance for example). I bet a large percentage of General Officers are Infantry or another Combat Arms. Best of Luck!

Infantry and/or being in a special forces unit.