I usually attack early if I have horses or elephants, but only to take territory I want; I stop after one or two cities. Then again and again every few hundred years.
I don't usually fatally harm the enemy, just chip away at them gradually, because I'm really only fighting so my empire can expand, not for the sake of fighting itself. I don't see the point of taking cities I don't want. Guess I'm not a warmonger at heart.
My present Russia game has gone like this:
310 BC - 10AD : destroyed one German city, took another
590 - 1060AD : destroyed a Dutch city, captured/destoyed 4 Byzantine cities and took 1 German city
1280 - 1340: took 2 Greek cities
1480 - 1600: took 2 Spanish cities
1764 - 1880: 6 Mongolian (this was more than usual because I wanted to build the trans-Siberian railway), 2 Roman, 1 Ottoman cities
1908 - 1928: 3 Austrian cities
This was fairly typical of games when I play a European power. I've had Austrian and Roman games follow the same pattern (even without the AI keeps building nice targets all around, as it does for Russia
). But when playing the Indians, for example, conquests were very few; it was mostly culture flips. And with Ethiopia I only beat up on Egypt and the Zulus.
Hope this helps.