You speak of balance, that is where the problem is.
While there needs to be a balance, the right balance is so far away from what would seem the right balance to you that probably civ looks like a game where balances don't work, but extremes do work.
And well, that is somewhat right.
In most cases, the balance is that you want just enough warriors early on to keep the barbs out, no more than that, no less.
Very often, when i have like 15 cities, the units i have are like 7-8 worriors and about 12-18 workers. That is the balance that works. That is enough workers to get your settlers to their position without being killed by barbs.
The AI is no threat to you, in early game, they won't just attack you, they will probably threaten you and make you pay them. Just pay them and have your revenge later. This is true on every difficulty level, but even moreso on the lower ones.
Start building an army when the space for expanding is running out. When there is no more place to go, that is the moment when the AI will start to get agressive.
Later in the game, the balance is also very one sided, but pretty much the other side around. If you want war, it is usually best to have 90%-100% of your cities producing military units, not half of them or something.
In civ, you need targets and then you go for that target with all you can.
In early game, your target is expansion. The only thing that counts is expansion.
After that, there is normally a buildup phase where you build some city improvements and let your cities grow in size. Most players have the tendency to build much, much to many city improvements though. On the lowest levels, you could do very well by not building any other improvements that barracks.
I advise you to do the following to get a different feel for civ.
Choose the iroquois tribe, play on a pangea map, medium size, maximum number of opponents and sedentary barbs.
Now, start your game with 1 warrior only, send this warrior around the world to explore. Meanwhile, build only settlers, workers and a few granaries in your best cities. Expand as fast as you possibly can.
When you have 10-12 cities, stop expanding completely, unless you need another city to get horses. The year 1000BC is your goal to have these 10-12 cities. If you have a food bonus next to your capital, that is a pretty easy goal, if you don't have a food bonus, it is a somewhat high goal for a new player and you could be satisfied by having 9 cities at 1000BC. At this point, you are also expected to have about as many workers as you have cities. warriors are of no importance with your sedentary barbs, build them only when your cities don't have the population to build settlers or workers.
Now, build barracks in every city. Now build mounted warriors from all your cities and start attacking as soon as you have 8 of them. Take your oponents down one by one.
Maybe you want to build some aquaducts in cities to grow them past size 6, maybe you will have won before your cities need any aquaducts.
It might sound impossible, but you can very well win in ancient age this way, provided you have horses that is. As long as your opponents don't have pikemen, you will crush them. If they do get pikemen, step back, stop conquest, buildup your cities to let them grow past 6, maybe with a library and/or market in some of them. Then continu your conquest when you have chivalry. If you do play well and you really focus on growth first, then conquest, you wont need the middle ages.
Just do that, and you will find another way to play civ. I can tell you firsthand that what you experience in this sample game is a very good feel of how to play and win civ, even on the highest difficulty levels.
After you have done that, go play around with other things and think of strategies and game plans for your own.