At what point does the main quest actually start? I'm about to start a new character and wanted to play as someone who travels from town to town making her way be helping others out. I'd like to do this in a Skyrim without dragons, so is this possible by not letting the main quest start?

The main quest automatically starts however Dragons do not appear until you complete the quest Dragon Rising.

Technically speaking the main quest starts as soon as you click play. Just ton't talk to the Jarl of Whiterun and you will be dragon free.

playing without Dragons can be done by:

-Not starting Dragon Rising Main Quest

-Finishing The Horn of Jurgen Windcaller & not doing A Blade In The Park

-There's also a mod that lets you start off however you want and not even be a dragonborn on Nexus mod

I'm not so much a fan of having dragons ramdomly showing up when I'm overencumbered & travelling between towns to sell items either. So I normally go with the 2nd route (not doing A Blade In The Dark). This option would allow the player to get the full Fus Ro Dah shout and open the option of learning other Shouts & kill Dragons "at their Lairs". All that needs to be done is you just ignore Delphine right after she returns the Horn to you and asked that you follow her into a room (You can follow her, jump in front of her at the stairs, then quicky loot all the free items in that room and run out before she starts her dialogue and never return there again). And remember, when you tried to talk to her in Whiterun when she came to see the Arl's wizard, she was saying "Leave me alone", so you can just say that to her this time.

I never heard of that quest! What kind of park? A water park? I hope it's a blade in a water park!

Simply never visit Bleak Falls Barrow. If you roleplay then you may want to talk to the jarl of Whiterun, but your character could decide that the whole issue with dragons is something only suicidal fools would want to solve.

Not sure if you are on PC or console, but Skyrim Unbound let's you play without any Dragons, Dragons only at word walls, and even as a non-Dragonborn character.