Tips for Civ III warfare:
1) Mobile units
2) Overwhelming force
1 is easy. Build horsemen, make 'em knights once you have Chivalry, and then make them Cavalry once you have Mil. Tradition. Lots and lots of Cavalry. Pre-infantry, that's all you need. You can often beat the AI to Military Tradition, which also usually means they don't have nationalism (riflemen). Cavalry slaughter musketmen.
2 can be more difficult, but can be attain either by build tons and tons of units yourself, or building as much as you can and enlisting an ally or allies in the war you wish to fight. Often, the best scenario is joining a war in progress between two neighbors. Pick a side and pounce - and make an alliance w/the other side, because then they will LOVE you.
I did this last night. The Iroquois attacked Rome. Mounted Warriors fighting Legionaries... it looked like a bloodbath, with the Iroquois eventually winning. First off, Rome and I (Egypt) were quite friendly, and I felt the Iroquois were getting too powerful. Plus, they had the Middle East (world map), and I wanted it. Finally, I was building Sun Tzu and the Sistine, but so were some others, so a Golden Age was just what I needed. My Knights
wasted their MW's, and even when they managed to build some knights, it was too little, too late. I now own all of Asia and the northern 1/2 of Africa. The last Iroquois city is in the Azores, off the coast of Portugal. Rome gained nothing.
-Arrian