Infantry regiments are the basic unit of your military. Along with being the quickest unit to recruit, they are also the cheapest to purchase and maintain. These units will perform better when defending in rugged terrain and worse when attacking in open terrain. Infantry are essential to any siege if you wish to make an assault.
Unlike cavalry and artillery, infantry tends to have balanced effectiveness in both shock and fire, although the fire modifier is 0 until Land Tech 17 and isn't really meaningful until it reaches .5 at Land Tech 22. Infantry units generally have more dots than artillery and cavalry units at the same respective level, but lower damage modifiers - this makes them generally better at defense and morale than the other respective units, while doing less damage.
Every infantry unit has a maneuver value of 1. The developers tried to share certain units like the Eastern Carabiniers between the Ottoman and Eastern techgroup, which can be seen from the text files. (The lines unit_type = eastern; unit_type = ottoman can be found in these text files) However, for such units, they are only available to the Ottoman techgroup, when played in the game.
Cavalry regiments at the start of the game will be far superior than infantry. As time and technology progresses infantry will close this gap until cavalry only have a minor advantage in combat. As of Heir to the Throne, cavalry have the same movement speed as infantry, whereas previously they were faster. They are more expensive than infantry to purchase and maintain, take longer to recruit and are essentially useless in an assault.
Cavalry have the best shock modifiers in the game, but are generally useless at delivering fire damage. Moreover, they tend to have relatively low morale and defense values compared to infantry. This makes them able to deliver a punishing first strike (as shock is the first phase), but leaves them vulnerable to being short on morale or being slaughtered in the fire phase - especially from about Land Tech 33 on. However, armies that contain cavalry gain a 'flanking' bonus for up to 4 regiments of cavalry during combat resolution that will greatly improve the damage done by your army. Therefore it is advisable to have around 4 or 5 regiments of cavalry in your army.
Every cavalry unit has a maneuver value of 2. The developers tried to share certain units like the Eastern Skirmishers and Eastern Uhlans between the Ottoman and Eastern techgroup, which can be seen from the text files. (The lines unit_type = eastern; unit_type = ottoman can be found in these text files) However, for such units, they are only available to the Ottoman techgroup, when played in the game.
Artillery regiments represent a smaller infantry regiment with artillery support. For most countries, artillery will become available once that country reaches land technology level 12. At first artillery will be most effective at greatly reducing the time needed for sieges. With refinements to cannons through greater technology they will become weapons that are quite effective in the field as well. They are very slow moving and very expensive to purchase and maintain.
IN Onwards: Artillery can fight from the back row, doing half damage and lending half its defense to the unit in front of it. In doing so, you can essentially deliver more combat power and staying power in a shorter line. While artillery will never have a very powerful shock phase, the fire phase reaches a modifier of 2 at Land Tech 33, 3 at 35, 4 at 39, 5, at 52, 6 at 59, and 8 at 61 - at level 61, an artillery unit in the back row has nearly the same fire modifier as a cavalry unit in the front row!