You are asking a very broad question, we're talking about the changes in the whole society over 1200 years. It is the equivalent of asking about French history from Charlemagne to the present day.

Further, the division of Kingdom/Republic/Empire doesn't really work. For instance, the Western Empire of the 5th century, ravaged by centuries of civil wars, with the puppet Emperors and generals holding all the power, an impotent Senate, and the city of Rome itself becoming all but irrelevant is so different from the Empire of Augustus that it only shares the name.

As /u/mmilosh said, this is an extremely broad question. I can outline a handful of things (primarily on the military side) for you though, with the disclaimer that I'm answering in broad strokes to a broad question.

1) Kingdom of Rome

The issue with the kingdom period is that we don't know a whole lot about it. Perhaps someone with more knowledge on archeology from the period can chime in on this, but honestly, I don't think from the source material (Livy and Virgil primarily, in my experience) we can give a definitive answer on effectiveness or ineffectiveness. It was a local kingdom or city-state amongst large groups of other such kingdoms and city states.

2) The Republic

The Republic was both effective and ineffective, as most nations are in some regards. The effectiveness of a (generally) non professional military in conquering the Italian peninsula, Cisalpine Gaul, Carthage, and Greece is fascinating to study. Once the army becomes more professional, you have the conquests of Gaul and the solidification of power around the western Mediterranean and the movement east. This was, to that point, a golden period in Roman history with extensive control of land, trade routes, and security within and without.

3) The Empire

Now the empire, generally, is divided into two sections. The Principate (Augustus-Carinus, or up to the reign of Diocletian) and the Dominate (Diocletian-Romulus Augustulus, nominally). I would argue based on the source material (Cassius Dio, Tacitus, Josephus, Pliny, among others) and the secondary research that we have (extensive amounts, particularly from Adrian Goldsworthy in the recent as it pertains to Roman military) that the Principate represents the height of effectiveness in the realm of military. I'm sure it could also be argued that socially and economically Rome was at it's height at this point in time, given control of the Mediterranean and partial control of trade routes coming from the east and the continued on and off conflict with the Persian empires in the east.

The Dominate period, while having many bright spots, is also a period of decline towards the eventual "de-Romanization" of Rome through the influx of foreign tribes and the shift of focus to the east, setting up the Byzantine empire (or Eastern Roman Empire, Romaioi, for students of Byzantine history.) So I would argue that while the empire is effective in this period at certain times in defending itself, the general effectiveness begins to decline.

Now, this is a very broad stroke through a millennium of history. If you have any specific questions, I would be happy to try and answer!