I agree with the op and I am in the same situation as you. I recently started playing civ III again and I find it is the most addicting game there is. For the first time in a while I do not want to turn the game off. When I hit quit I feel bad to say "Yes, Immediately!" because its a lie. I need to go to bed sometime before 3 am more often. But I have also seen the game is a lot harder. In the first file I played I went as Russia (unaware that they are so awful) quickly invaded a neighbor (Monte) and then got my ass handed to me when my invasion force was killed and a few jaguars were parked outside Moscow. I didn't stick around for the last turn when Moscow would be burned to the ground my vicious Aztec Jaguars. Now I am going as Japan and having doing a lot better as I have grasped a little bit more of the strategy. I'm only on chieftain difficulty.
If you're Russia, as vmxa said, they're Expansionistic. On Chieftain that's actually a strong trait. With a couple of scouts exploring, tribal villages will get you most of the AA techs. With Scientific you should always have a tech lead. Even with Japan, Chieftain is where the villages are very nice to you.
Yes, the game is addicting. It also feels faster paced than IV.
Siege and Armies make war a lot easier. Getting that first army really makes the invasions easier. And the AI doesn't use armies at all well. And with Jaguars and Militaristic Monty isn't an ideal early war target. Before an invasion I'd check the Military advisor and make sure you have more power than the target does. With good tactics you can overcome a stronger foe but for now, just be sure you're stronger.