I think it can be said that you will get into war with someone at some point. This isn't catastrophical as long as you are adequately prepared.
Early DoWs (declaration of war) are a possibility escpecially if you have aggressive neighbours nearby, such as montezuma or alexander or shaka, those are the kind of nutjubs that you would do well to rush before they get too strong.
I'm sure that others who know more of the diplomacy side of civ4 could give you better tips, but the main deal with diplomacy is that you cannot REALLY please everyone.
When you bow towards a certain civ, you're showing your backside to the other
To some extent that's true, A.I. civs have a "worst enemy" and should you interact too much with them (the worst enemy), you will get minus points to your relations with the first civ.
Also it might be good not to be very prudish about religion in the game, oftentimes if you're a "heretic" lets say that the entire civ4 world is either buddhist or hindu, then it's probably not a good idea to be taoist.
As to the size of your military, there are the demographics somewhere around the upper right corner of the screen and there you'll find you're power slider. Its relative to the strength of other know civs I think. If you are not the most "powerful" then it's not catastrophical either, but you should definitely try to keep somewhere near the top3 at least. If you are like the worst in the power rating out of all the civs then you're definitely gonna run into some problems.
Once the warmongers: Alex Shaka Monty see weakness, they go for the kill usually. Also you shouldn't for that reason leave you're border cities weakly defended I think. After early game and astronomy the threat of naval invasions also materializes so coastal cities should get some protection also. It gets trickier if you're right in the middle of the continent of course, but generally you should strongly defend your borders and have a reserve attack force ready to deploy if you get DoWed. Counter-attack and crush the enemy armies, retake possibly lost land's and perhaps even take a few enemy cities for revenge, and try to sue for peace.
War doesn't always have to be a "war of annihilation" as it were, total war to the unconditional surrender. Especially if you feel that waging a big war for a long time against a certain civ might cause other civs to join in while you're still at it.
You should be very careful however before you give strong military techs away as bribes (such as rifling/steel/assembly line etc...)
Also bribing A.I.s into attacking each other is excellent diplomacy, with money and/or techs. If you can do it only with money, then you can be satisfied since the both war waging A.I.s will slow down their tech in favour of waging war and producing units. You might also consider dogpiling a war-waging A.I. yourself, it sometimes yields excellent results, oftentimes the A.I.s cities are very poorly defended, except maybe for the capital, as most of their units would be fighting and enemy elsewhere.