To start the Civil War quest, just escape from Helgan. As you're running out of the starting caves, X/Y soldier will tell you to join X/Y side's on the civil war and report to Tullius/Ulfric. That's what starts it. Whatever you do, just ignore Balgruuf or keep the quest his court mage offers permanently, and no dargon shall spawn.
EDIT: nvm, goes it show that I need to replay SK again.
I am curious as to what would happen if you begin Dragon Rising but attempt to avoid killing the Dragon and not finishing it. This should be in hints section technically, cause here comes a spoiler. I am not sure it will begin the Battle for Whiterun properly if you don't finish Dragon Rising. Anyway, let us know how that turns out. I am pretty sure Baalgruf won't talk to you about the Axe part until you kill the dragon at the watchtower.
you are mistaken...
in either side of the Civil War Questline you have to talk to Balgruuf before te battle of Whiterun (if your an Imperial yo offer him your support, if a stormcloak to deliver Ulfrics Axe).
If you have not dealt with the Dragon at the watch tower Balgruuf will say "No time for that! go kill that dragon, then we can talk about it!".
It's especially annoying when you want to be a stormcloak.
Balgruuf- "No, I won't declare war on you until you help me kill the dragon!"
You-
The interesting thing here is, if you wait with doing Dragon Rising until Balgruuf forces you, he comes off as much less likeable, especially if you play the SC side. I've done this with one character, and my reaction was pretty much "why force me to help me with that dragon when you KNOW I'm here with a message from Ulfric, whom you have no intention of joining anyway, and then you have the stomach to name me Thane, and act SURPRISED when I later show up as your enemy in battle?!?" He just seems demanding and selfish when you do the quests in that order. (I supposed he still would seem likeable if you come to him as a legionnaire, however, since you then will remain on the same side.)
@ the OP: If you're on PC I'm pretty sure there are mods that allow you to play through the CW without spawning dragons.
I'm RPing my Redguard as the non-Dragonborn harbinger of the Companions. He helped kill Mira-whatever at the watchtower, then scat before it was totally dead, ergo no soul absorption. So now there's a dragon corpse rotting in the middle of Skyrim. No more dragons will show up, and my character is free to go about his business: smith armor, drink mead, play tag with his two kids, and go on the occasional side quest.
My Nord never even bothered to go to Bleak Falls Barrow, since combat isn't really her specialty.