What quest triggers the random dragon attacks?

Dragons in Skyrim don't start appearing as random encounters until a certain point in the main quest - I'm not ready to take on the main quest, but I do want dragons about so I can unlock the words of power I'm finding.

What causes the Dragons to start spawning?

Does additional progress along the Main Quest affect the chance of dragons spawning?

Once you kill the first dragon for the main quest, random dragons can start spawning. It's the second quest in the main questline, so it won't take you very long at all, and it won't reveal too much about where the main quest is going.

Specifically, the quest you need to complete is called "Dragon Rising".

It is Dragon's Rising like others have said. Here's a tip: Before doing Dragon's Rising, go to the Word Walls and collect all of the words of power. There won't be dragons yet so it should be relatively easy. Then, go do Dragon's Rising. Now, for the rest of the game, whenever you kill a dragon you can unlock a new word since you already have most of the words.