In Civ2, I used to play the Americans a lot. I still play them now, and their relatively "weak" abilities make them an interesting challenge. In fact, I find that with a bit of luck in the early game, the Americans can get a leg up in the tech race and have a good head start on the early-game Wonders. I used to think that the Industrious ability was pretty lame, until I realized that I was under-using my Workers. A good crew of fast Workers and judicious land management can make you a powerhouse.
(My #1 tip for land management: don't waste food. If your city has hit the 6 or 12 population barrier, try to shuffle the laborers around so that you are producing maximum shields and/or trade, with just enough food to keep it at zero growth level until you get that Aqueduct or Hospital.)
As far as the "Most Annoying Neighbors" prize goes, since I'm usually the Americans, it's the Aztecs who tick me off the most. I prefer to drive for Iron Working and Horeseback Riding early in the game, then go on a jaguar (warrior) hunt. If I can't cut Montezuma down to size in the Iron Age, then I wait till the medieval weapons come into play. Nothing slices through the Aztec empire like a good combined-arms force of Longbowmen and Catapults escorted by Pikemen, with a few Knights to mow down the stragglers.
So the Aztecs are rarely a force worth worrying about after the early Middle Ages. My next most hated/feared opponent is the Persians. Their armies of Immortals ensure that they won't get taken down early, and they tend to build up a strong science base and become annoyingly competitive by the Industrial Age. If at all possible, I try to get a Mutual Defense Pact with one or more of the Persians' neighbors, and take them on in the age of Riflemen and Cavalry, when their Immortals have lost their edge.
My second favorite civ to play in Civ3 is the Germans, and their worst neighbor is the Russians. Scientific to compete with me in the tech race, and Expansionist to grab all the goodies in the early game! Ugh! I try to take advantage of German Militarism and hit them early in the game with a few well-placed Archers and Warriors. This checks their expansion and, if I'm really lucky, gets me a Great Leader and the Heroic Epic in the ancient era.
Back in Civ2, I had a fairly long list of civs that I would attack as soon as possible. Mongols were aggressive and the only form of persuasion they understood was cold steel. Greeks were just too darn competitive to allow to survive. Zulus, well, everyone knows it's not a question of whether the Zulus will attack you, but when, so do unto them first. And Indians and French -- well, I just find it fun to kill Indians and French.